The idea of interference alignment Let h 1 i y 1 . . . . H i = and Y = . . h Bi y B Then the system is Y = x 1 H 1 + ( x 2 H 2 + · · · + x K H K ) � �� � ∈ Span { H 2 ,..., H K } If H 1 �∈ Span { H 2 , . . . , H K } then x 1 can be uniquely determined from the above equations. Cadambe & Jafar (2008) showed that it is theoretically possible that each of the K users enjoy half of the available signalling dimensions. More formally, DOF = K 2 . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 4 / 24
The idea of interference alignment Let h 1 i y 1 . . . . H i = and Y = . . h Bi y B Then the system is Y = x 1 H 1 + ( x 2 H 2 + · · · + x K H K ) � �� � ∈ Span { H 2 ,..., H K } If H 1 �∈ Span { H 2 , . . . , H K } then x 1 can be uniquely determined from the above equations. Cadambe & Jafar (2008) showed that it is theoretically possible that each of the K users enjoy half of the available signalling dimensions. More formally, DOF = K 2 . Various assumptions: - varying channel coefficients, - multiple antennae at receivers - diagonal form of the channel coefficient matrices - . . . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 4 / 24
Rational dimensions framework V. Beresnevich (University of York) Diophantine approximation 18 January 2017 5 / 24
Rational dimensions framework Assume the transmit signals are taken from the constellation { 0 , . . . , Q } . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 5 / 24
Rational dimensions framework Assume the transmit signals are taken from the constellation { 0 , . . . , Q } . By observing only y = h 1 x 1 + · · · + h K x K at the receiver we still can decode our signals, if h 1 , . . . , h K are linearly independent over Q (or equivalently Z ). V. Beresnevich (University of York) Diophantine approximation 18 January 2017 5 / 24
Rational dimensions framework Assume the transmit signals are taken from the constellation { 0 , . . . , Q } . By observing only y = h 1 x 1 + · · · + h K x K at the receiver we still can decode our signals, if h 1 , . . . , h K are linearly independent over Q (or equivalently Z ). Independence over Q means that any linear combination h 1 x 1 + · · · + h K x K with integer x 1 , . . . , x K will only vanish when all the x i are zeros. V. Beresnevich (University of York) Diophantine approximation 18 January 2017 5 / 24
Rational dimensions framework Assume the transmit signals are taken from the constellation { 0 , . . . , Q } . By observing only y = h 1 x 1 + · · · + h K x K at the receiver we still can decode our signals, if h 1 , . . . , h K are linearly independent over Q (or equivalently Z ). Independence over Q means that any linear combination h 1 x 1 + · · · + h K x K with integer x 1 , . . . , x K will only vanish when all the x i are zeros. If there is noise z , the receiver observes y = h 1 x 1 + · · · + h K x K + z . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 5 / 24
Rational dimensions framework Assume the transmit signals are taken from the constellation { 0 , . . . , Q } . By observing only y = h 1 x 1 + · · · + h K x K at the receiver we still can decode our signals, if h 1 , . . . , h K are linearly independent over Q (or equivalently Z ). Independence over Q means that any linear combination h 1 x 1 + · · · + h K x K with integer x 1 , . . . , x K will only vanish when all the x i are zeros. If there is noise z , the receiver observes y = h 1 x 1 + · · · + h K x K + z . This observation can be distinguished from another one, say y ′ = h 1 x ′ 1 + · · · + h K x ′ K + z ′ only if y and y ′ are sufficiently separated. V. Beresnevich (University of York) Diophantine approximation 18 January 2017 5 / 24
Rational dimensions framework Assume the transmit signals are taken from the constellation { 0 , . . . , Q } . By observing only y = h 1 x 1 + · · · + h K x K at the receiver we still can decode our signals, if h 1 , . . . , h K are linearly independent over Q (or equivalently Z ). Independence over Q means that any linear combination h 1 x 1 + · · · + h K x K with integer x 1 , . . . , x K will only vanish when all the x i are zeros. If there is noise z , the receiver observes y = h 1 x 1 + · · · + h K x K + z . This observation can be distinguished from another one, say y ′ = h 1 x ′ 1 + · · · + h K x ′ K + z ′ only if y and y ′ are sufficiently separated. Fact: for any ε > 0 for almost every collection ( h 1 , . . . , h K ) of real numbers there exists a constant γ = γ ( ε, h 1 , . . . , h K ) > 0 such that γ | h 1 ( x 1 − x ′ 1 ) + · · · + h K ( x K − x ′ K ) | ≥ Q K − 1+ ε whenever ( x 1 , . . . , x K ) � = ( x ′ 1 , . . . , x ′ V. Beresnevich (University of York) Diophantine approximation 18 January 2017 5 / 24 K ).
Rational dimensions: single antenna, multiple data streams Using that Fact a single antenna can be turned into a multiple antennae to simultaneously transmit several data streams, say x 1 , . . . , x M , x i ∈ { 0 , . . . , Q } . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 6 / 24
Rational dimensions: single antenna, multiple data streams Using that Fact a single antenna can be turned into a multiple antennae to simultaneously transmit several data streams, say x 1 , . . . , x M , x i ∈ { 0 , . . . , Q } . The transmit signal may be taken to be x = λ ( c 1 x 1 + · · · + c M x M ) where | c i | = O (1) are precoding coefficients and λ is a scalar reflecting power constraints. V. Beresnevich (University of York) Diophantine approximation 18 January 2017 6 / 24
Rational dimensions: single antenna, multiple data streams Using that Fact a single antenna can be turned into a multiple antennae to simultaneously transmit several data streams, say x 1 , . . . , x M , x i ∈ { 0 , . . . , Q } . The transmit signal may be taken to be x = λ ( c 1 x 1 + · · · + c M x M ) where | c i | = O (1) are precoding coefficients and λ is a scalar reflecting power constraints. The separation between different x ’s is at least λ O ( Q − M +1 − ε ). V. Beresnevich (University of York) Diophantine approximation 18 January 2017 6 / 24
Rational dimensions: single antenna, multiple data streams Using that Fact a single antenna can be turned into a multiple antennae to simultaneously transmit several data streams, say x 1 , . . . , x M , x i ∈ { 0 , . . . , Q } . The transmit signal may be taken to be x = λ ( c 1 x 1 + · · · + c M x M ) where | c i | = O (1) are precoding coefficients and λ is a scalar reflecting power constraints. The separation between different x ’s is at least λ O ( Q − M +1 − ε ). Choosing 1 − ε 2( M + ε ) ) λ = P 1 / 2 Q − 1 Q = O ( P and ensures that the separation is bigger than the standard deviation of the noise, while meeting power constraints. V. Beresnevich (University of York) Diophantine approximation 18 January 2017 6 / 24
Rational dimensions: single antenna, multiple data streams Using that Fact a single antenna can be turned into a multiple antennae to simultaneously transmit several data streams, say x 1 , . . . , x M , x i ∈ { 0 , . . . , Q } . The transmit signal may be taken to be x = λ ( c 1 x 1 + · · · + c M x M ) where | c i | = O (1) are precoding coefficients and λ is a scalar reflecting power constraints. The separation between different x ’s is at least λ O ( Q − M +1 − ε ). Choosing 1 − ε 2( M + ε ) ) λ = P 1 / 2 Q − 1 Q = O ( P and ensures that the separation is bigger than the standard deviation of the noise, while meeting power constraints. More sophisticated examples require separability of linear forms when the coefficients the forms are functions of several other variables (in DA this is known as Diophantine approximation on manifolds). V. Beresnevich (University of York) Diophantine approximation 18 January 2017 6 / 24
Rational dimensions: single antenna, multiple data streams Using that Fact a single antenna can be turned into a multiple antennae to simultaneously transmit several data streams, say x 1 , . . . , x M , x i ∈ { 0 , . . . , Q } . The transmit signal may be taken to be x = λ ( c 1 x 1 + · · · + c M x M ) where | c i | = O (1) are precoding coefficients and λ is a scalar reflecting power constraints. The separation between different x ’s is at least λ O ( Q − M +1 − ε ). Choosing 1 − ε 2( M + ε ) ) λ = P 1 / 2 Q − 1 Q = O ( P and ensures that the separation is bigger than the standard deviation of the noise, while meeting power constraints. More sophisticated examples require separability of linear forms when the coefficients the forms are functions of several other variables (in DA this is known as Diophantine approximation on manifolds). Just as with vector alignment, it is possible to align alone real numbers. V. Beresnevich (University of York) Diophantine approximation 18 January 2017 6 / 24
2-user X-channel (Motahari, et al ) V. Beresnevich (University of York) Diophantine approximation 18 January 2017 7 / 24
2-user X-channel (Motahari, et al ) h 11 T 1 ✲ R 1 u 1 , u 2 ? u 1 , v 1 x 1 y 1 ❅ � ✒ ❅ � ❅ � h 12 ❅ � � ❅ h 21 � ❅ � ❅ � ❘ ❅ T 2 ✲ R 2 u 2 , v 2 x 2 y 2 v 1 , v 2 ? h 22 V. Beresnevich (University of York) Diophantine approximation 18 January 2017 7 / 24
2-user X-channel (Motahari, et al ) h 11 T 1 ✲ R 1 u 1 , u 2 ? u 1 , v 1 x 1 y 1 ❅ � ✒ ❅ � ❅ � h 12 ❅ � � ❅ h 21 � ❅ � ❅ � ❘ ❅ T 2 ✲ R 2 u 2 , v 2 x 2 y 2 v 1 , v 2 ? h 22 T 1 simultaneously transmits two data streams u 1 (intended for R 1 ) and v 1 (intended for R 2 ). Similarly, T 2 transmits independent two data streams u 2 (intended for R 1 ) and v 2 (intended for R 2 ). h ij are the the channel coefficients. Let x i is the signal transmitted by T i . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 7 / 24
2-user X-channel (Motahari, et al ) h 11 T 1 ✲ R 1 u 1 , u 2 ? u 1 , v 1 x 1 y 1 ❅ � ✒ ❅ � ❅ � h 12 ❅ � � ❅ h 21 � ❅ � ❅ � ❅ ❘ T 2 ✲ R 2 u 2 , v 2 x 2 y 2 v 1 , v 2 ? h 22 T 1 simultaneously transmits two data streams u 1 (intended for R 1 ) and v 1 (intended for R 2 ). Similarly, T 2 transmits independent two data streams u 2 (intended for R 1 ) and v 2 (intended for R 2 ). h ij are the the channel coefficients. Let x i is the signal transmitted by T i . The received signals by R 1 and R 2 are = h 11 x 1 + h 12 x 2 + z 1 , y 1 y 2 = h 21 x 1 + h 22 x 2 + z 2 . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 7 / 24
2-user X-channel (Motahari, et al ) y 1 = h 11 x 1 + h 12 x 2 + z 1 , y 2 = h 21 x 1 + h 22 x 2 + z 2 . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 8 / 24
2-user X-channel (Motahari, et al ) y 1 = h 11 x 1 + h 12 x 2 + z 1 , y 2 = h 21 x 1 + h 22 x 2 + z 2 . If x 1 = λ ( h 22 u 1 + h 12 v 1 ) and x 2 = λ ( h 21 u 2 + h 11 v 2 ) then � � y 1 = λ h 11 h 22 u 1 + h 12 h 21 u 2 + h 11 h 12 ( v 1 + v 2 ) + z 1 , � � y 2 = λ h 21 h 22 ( u 1 + u 2 ) + h 12 h 21 v 1 + h 11 h 22 v 2 + z 1 At R 1 : v 1 and v 2 are aligned along the same real number, h 11 h 12 At R 2 : u 1 and u 2 are aligned along the same real number, h 21 h 22 V. Beresnevich (University of York) Diophantine approximation 18 January 2017 8 / 24
2-user X-channel (Motahari, et al ) y 1 = h 11 x 1 + h 12 x 2 + z 1 , y 2 = h 21 x 1 + h 22 x 2 + z 2 . If x 1 = λ ( h 22 u 1 + h 12 v 1 ) and x 2 = λ ( h 21 u 2 + h 11 v 2 ) then � � y 1 = λ h 11 h 22 u 1 + h 12 h 21 u 2 + h 11 h 12 ( v 1 + v 2 ) + z 1 , � � y 2 = λ h 21 h 22 ( u 1 + u 2 ) + h 12 h 21 v 1 + h 11 h 22 v 2 + z 1 At R 1 : v 1 and v 2 are aligned along the same real number, h 11 h 12 At R 2 : u 1 and u 2 are aligned along the same real number, h 21 h 22 Again, assuming u i , v i ∈ { 0 , . . . , Q } we achieve the required separation (and normalising power) in each of the equation by taking 1 − ε 1+ ε 2(3+ ε ) ) , 3+ ε ) . Q = O ( P λ = O ( P DOF= 4 3 almost surely. V. Beresnevich (University of York) Diophantine approximation 18 January 2017 8 / 24
Diophantine approximation: the basics In applications we often deal with small linear forms or systems of small linear forms at integer variables - V. Beresnevich (University of York) Diophantine approximation 18 January 2017 9 / 24
Diophantine approximation: the basics In applications we often deal with small linear forms or systems of small linear forms at integer variables - a subject of Diophantine approximation. V. Beresnevich (University of York) Diophantine approximation 18 January 2017 9 / 24
Diophantine approximation: the basics In applications we often deal with small linear forms or systems of small linear forms at integer variables - a subject of Diophantine approximation. Dirichlet’s theorem: Let α i , j ∈ R , where 1 ≤ j ≤ n , 1 ≤ i ≤ m , and Q > 1. Then there exist q 1 , . . . , q n , p 1 , . . . , p m ∈ Z such that | q 1 α i , 1 + · · · + q n α i , n − p i | < Q − n (1 ≤ i ≤ m ) m 1 ≤ max 1 ≤ j ≤ n | q j | ≤ Q . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 9 / 24
Diophantine approximation: the basics In applications we often deal with small linear forms or systems of small linear forms at integer variables - a subject of Diophantine approximation. Dirichlet’s theorem: Let α i , j ∈ R , where 1 ≤ j ≤ n , 1 ≤ i ≤ m , and Q > 1. Then there exist q 1 , . . . , q n , p 1 , . . . , p m ∈ Z such that | q 1 α i , 1 + · · · + q n α i , n − p i | < Q − n (1 ≤ i ≤ m ) m 1 ≤ max 1 ≤ j ≤ n | q j | ≤ Q . Minkowski’s theorem for systems of linear forms: Let β i , j ∈ R , where 1 ≤ i , j ≤ k , and let C 1 , . . . , C k > 0. If k � | det( β i , j ) 1 ≤ i , j ≤ k | ≤ C i , (1) i =1 then there exist a non-zero integer point x = ( x 1 , . . . , x k ) such that � | x 1 β i , 1 + · · · + x k β i , k | < C i , (1 ≤ i ≤ k − 1) (2) | x 1 β k , 1 + · · · + x n β k , k | ≤ C k . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 9 / 24
Diophantine approximation: the basics In applications we often deal with small linear forms or systems of small linear forms at integer variables - a subject of Diophantine approximation. Dirichlet’s theorem: Let α i , j ∈ R , where 1 ≤ j ≤ n , 1 ≤ i ≤ m , and Q > 1. Then there exist q 1 , . . . , q n , p 1 , . . . , p m ∈ Z such that | q 1 α i , 1 + · · · + q n α i , n − p i | < Q − n (1 ≤ i ≤ m ) m 1 ≤ max 1 ≤ j ≤ n | q j | ≤ Q . Minkowski’s theorem for systems of linear forms: Let β i , j ∈ R , where 1 ≤ i , j ≤ k , and let C 1 , . . . , C k > 0. If k � | det( β i , j ) 1 ≤ i , j ≤ k | ≤ C i , (1) i =1 then there exist a non-zero integer point x = ( x 1 , . . . , x k ) such that � | x 1 β i , 1 + · · · + x k β i , k | < C i , (1 ≤ i ≤ k − 1) (2) | x 1 β k , 1 + · · · + x n β k , k | ≤ C k . Proof: uses Minkowski’s theorem for convex bodies. V. Beresnevich (University of York) Diophantine approximation 18 January 2017 9 / 24
Diophantine approximation: the basics Minkowski’s theorem: If B is a convex body in R n symmetric about 0 and vol B > 2 n then B contains a non-zero integer point. V. Beresnevich (University of York) Diophantine approximation 18 January 2017 10 / 24
Diophantine approximation: the basics Minkowski’s theorem: If B is a convex body in R n symmetric about 0 and vol B > 2 n then B contains a non-zero integer point. Minkowski can be used beyond the real case. Example: Dirichlet’s theorem for C (one dimensional): For any z ∈ C and any Q > 1 there exist p , q ∈ Z [ i ] = { a + bi : a , b ∈ Z } ( i = √− 1) such that 4 | qz − p | < π Q , 1 ≤ | q | ≤ Q . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 10 / 24
Diophantine approximation: the basics Minkowski’s theorem: If B is a convex body in R n symmetric about 0 and vol B > 2 n then B contains a non-zero integer point. Minkowski can be used beyond the real case. Example: Dirichlet’s theorem for C (one dimensional): For any z ∈ C and any Q > 1 there exist p , q ∈ Z [ i ] = { a + bi : a , b ∈ Z } ( i = √− 1) such that 4 | qz − p | < π Q , 1 ≤ | q | ≤ Q . Badly approximable systems/matrices: Let α i , j ∈ R , where 1 ≤ j ≤ n , 1 ≤ i ≤ m , and Q > 1. Then A = ( α i , j ) i , j is badly approximable if there exist c > 0 such that for all Q > 1 the only integer solution ( q 1 , . . . , q n , p 1 , . . . , p m ) to the system | q 1 α i , 1 + · · · + q n α i , n − p i | < cQ − n (1 ≤ i ≤ m ) m | q i | ≤ Q (1 ≤ j ≤ n ) is zero. V. Beresnevich (University of York) Diophantine approximation 18 January 2017 10 / 24
Diophantine approximation: the basics Minkowski’s theorem: If B is a convex body in R n symmetric about 0 and vol B > 2 n then B contains a non-zero integer point. Minkowski can be used beyond the real case. Example: Dirichlet’s theorem for C (one dimensional): For any z ∈ C and any Q > 1 there exist p , q ∈ Z [ i ] = { a + bi : a , b ∈ Z } ( i = √− 1) such that 4 | qz − p | < π Q , 1 ≤ | q | ≤ Q . Badly approximable systems/matrices: Let α i , j ∈ R , where 1 ≤ j ≤ n , 1 ≤ i ≤ m , and Q > 1. Then A = ( α i , j ) i , j is badly approximable if there exist c > 0 such that for all Q > 1 the only integer solution ( q 1 , . . . , q n , p 1 , . . . , p m ) to the system | q 1 α i , 1 + · · · + q n α i , n − p i | < cQ − n (1 ≤ i ≤ m ) m | q i | ≤ Q (1 ≤ j ≤ n ) is zero. Bad ( n , m ) is the set of badly approximable m × n matrices. V. Beresnevich (University of York) Diophantine approximation 18 January 2017 10 / 24
Weighted Diophantine approximation Dirichlet’s theorem with weights: Let α i , j ∈ R , where 1 ≤ j ≤ n , 1 ≤ i ≤ m . Let c = 1 and let r = ( r 1 , . . . , r n ), s = ( s 1 , . . . , s m ) be such that s j ≥ 0 , r i ≥ 0 , s 1 + · · · + s m = 1 , r 1 + · · · + r n = 1 . Then for any Q > 1 there exist q 1 , . . . , q n , p 1 , . . . , p m ∈ Z such that | q 1 α i , 1 + · · · + q n α i , n − p i | < c Q − s i (1 ≤ i ≤ m ) | q j | ≤ Q r j (1 ≤ j ≤ n ) ( q 1 , . . . , q n ) � = 0 . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 11 / 24
Weighted Diophantine approximation Dirichlet’s theorem with weights: Let α i , j ∈ R , where 1 ≤ j ≤ n , 1 ≤ i ≤ m . Let c = 1 and let r = ( r 1 , . . . , r n ), s = ( s 1 , . . . , s m ) be such that s j ≥ 0 , r i ≥ 0 , s 1 + · · · + s m = 1 , r 1 + · · · + r n = 1 . Then for any Q > 1 there exist q 1 , . . . , q n , p 1 , . . . , p m ∈ Z such that | q 1 α i , 1 + · · · + q n α i , n − p i | < c Q − s i (1 ≤ i ≤ m ) | q j | ≤ Q r j (1 ≤ j ≤ n ) ( q 1 , . . . , q n ) � = 0 . Weighted Badly approximable systems/matrices: Let α i , j ∈ R , where 1 ≤ j ≤ n , 1 ≤ i ≤ m , and Q > 1. Then A = ( α i , j ) i , j is ( r , s )-badly approximable if there exist c > 0 such that for all Q > 1 the only integer solution ( q 1 , . . . , q n , p 1 , . . . , p m ) to the above system is zero. V. Beresnevich (University of York) Diophantine approximation 18 January 2017 11 / 24
Weighted Diophantine approximation Dirichlet’s theorem with weights: Let α i , j ∈ R , where 1 ≤ j ≤ n , 1 ≤ i ≤ m . Let c = 1 and let r = ( r 1 , . . . , r n ), s = ( s 1 , . . . , s m ) be such that s j ≥ 0 , r i ≥ 0 , s 1 + · · · + s m = 1 , r 1 + · · · + r n = 1 . Then for any Q > 1 there exist q 1 , . . . , q n , p 1 , . . . , p m ∈ Z such that | q 1 α i , 1 + · · · + q n α i , n − p i | < c Q − s i (1 ≤ i ≤ m ) | q j | ≤ Q r j (1 ≤ j ≤ n ) ( q 1 , . . . , q n ) � = 0 . Weighted Badly approximable systems/matrices: Let α i , j ∈ R , where 1 ≤ j ≤ n , 1 ≤ i ≤ m , and Q > 1. Then A = ( α i , j ) i , j is ( r , s )-badly approximable if there exist c > 0 such that for all Q > 1 the only integer solution ( q 1 , . . . , q n , p 1 , . . . , p m ) to the above system is zero. Let Bad ( r , s ) be the set of ( r , s )-badly approximable m × n matrices. V. Beresnevich (University of York) Diophantine approximation 18 January 2017 11 / 24
Weighted Diophantine approximation Dirichlet’s theorem with weights: Let α i , j ∈ R , where 1 ≤ j ≤ n , 1 ≤ i ≤ m . Let c = 1 and let r = ( r 1 , . . . , r n ), s = ( s 1 , . . . , s m ) be such that s j ≥ 0 , r i ≥ 0 , s 1 + · · · + s m = 1 , r 1 + · · · + r n = 1 . Then for any Q > 1 there exist q 1 , . . . , q n , p 1 , . . . , p m ∈ Z such that | q 1 α i , 1 + · · · + q n α i , n − p i | < c Q − s i (1 ≤ i ≤ m ) | q j | ≤ Q r j (1 ≤ j ≤ n ) ( q 1 , . . . , q n ) � = 0 . Weighted Badly approximable systems/matrices: Let α i , j ∈ R , where 1 ≤ j ≤ n , 1 ≤ i ≤ m , and Q > 1. Then A = ( α i , j ) i , j is ( r , s )-badly approximable if there exist c > 0 such that for all Q > 1 the only integer solution ( q 1 , . . . , q n , p 1 , . . . , p m ) to the above system is zero. Let Bad ( r , s ) be the set of ( r , s )-badly approximable m × n matrices. Bad ( n , m ) = Bad (( 1 n , . . . , 1 n ) , ( 1 m , . . . , 1 m )). V. Beresnevich (University of York) Diophantine approximation 18 January 2017 11 / 24
Weighted Diophantine approximation Dirichlet’s theorem with weights: Let α i , j ∈ R , where 1 ≤ j ≤ n , 1 ≤ i ≤ m . Let c = 1 and let r = ( r 1 , . . . , r n ), s = ( s 1 , . . . , s m ) be such that s j ≥ 0 , r i ≥ 0 , s 1 + · · · + s m = 1 , r 1 + · · · + r n = 1 . Then for any Q > 1 there exist q 1 , . . . , q n , p 1 , . . . , p m ∈ Z such that | q 1 α i , 1 + · · · + q n α i , n − p i | < c Q − s i (1 ≤ i ≤ m ) | q j | ≤ Q r j (1 ≤ j ≤ n ) ( q 1 , . . . , q n ) � = 0 . Weighted Badly approximable systems/matrices: Let α i , j ∈ R , where 1 ≤ j ≤ n , 1 ≤ i ≤ m , and Q > 1. Then A = ( α i , j ) i , j is ( r , s )-badly approximable if there exist c > 0 such that for all Q > 1 the only integer solution ( q 1 , . . . , q n , p 1 , . . . , p m ) to the above system is zero. Let Bad ( r , s ) be the set of ( r , s )-badly approximable m × n matrices. Bad ( n , m ) = Bad (( 1 n , . . . , 1 n ) , ( 1 m , . . . , 1 m )). The transpose of Bad ( r , s ) is Bad ( s , r ). V. Beresnevich (University of York) Diophantine approximation 18 January 2017 11 / 24
Weighted Diophantine approximation Dirichlet’s theorem with weights: Let α i , j ∈ R , where 1 ≤ j ≤ n , 1 ≤ i ≤ m . Let c = 1 and let r = ( r 1 , . . . , r n ), s = ( s 1 , . . . , s m ) be such that s j ≥ 0 , r i ≥ 0 , s 1 + · · · + s m = 1 , r 1 + · · · + r n = 1 . Then for any Q > 1 there exist q 1 , . . . , q n , p 1 , . . . , p m ∈ Z such that | q 1 α i , 1 + · · · + q n α i , n − p i | < c Q − s i (1 ≤ i ≤ m ) | q j | ≤ Q r j (1 ≤ j ≤ n ) ( q 1 , . . . , q n ) � = 0 . Weighted Badly approximable systems/matrices: Let α i , j ∈ R , where 1 ≤ j ≤ n , 1 ≤ i ≤ m , and Q > 1. Then A = ( α i , j ) i , j is ( r , s )-badly approximable if there exist c > 0 such that for all Q > 1 the only integer solution ( q 1 , . . . , q n , p 1 , . . . , p m ) to the above system is zero. Let Bad ( r , s ) be the set of ( r , s )-badly approximable m × n matrices. Bad ( n , m ) = Bad (( 1 n , . . . , 1 n ) , ( 1 m , . . . , 1 m )). The transpose of Bad ( r , s ) is Bad ( s , r ). The transpose of Bad ( n , m ) equals Bad ( m , n ). V. Beresnevich (University of York) Diophantine approximation 18 January 2017 11 / 24
Multiplicative Diophantine approximation V. Beresnevich (University of York) Diophantine approximation 18 January 2017 12 / 24
Multiplicative Diophantine approximation Littlewood’s conjecture (LC), (1930): For any α 1 , α 2 ∈ R and any ε > 0 there exists q ∈ Z � =0 , p 1 , p 2 ∈ Z such that | q | · | q α 1 − p 1 | · | q α 2 − p 2 | < ε . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 12 / 24
Multiplicative Diophantine approximation Littlewood’s conjecture (LC), (1930): For any α 1 , α 2 ∈ R and any ε > 0 there exists q ∈ Z � =0 , p 1 , p 2 ∈ Z such that | q | · | q α 1 − p 1 | · | q α 2 − p 2 | < ε . What’s known: • V. Beresnevich (University of York) Diophantine approximation 18 January 2017 12 / 24
Multiplicative Diophantine approximation Littlewood’s conjecture (LC), (1930): For any α 1 , α 2 ∈ R and any ε > 0 there exists q ∈ Z � =0 , p 1 , p 2 ∈ Z such that | q | · | q α 1 − p 1 | · | q α 2 − p 2 | < ε . What’s known: • LC holds if either α 1 or α 2 is not badly approximable. • V. Beresnevich (University of York) Diophantine approximation 18 January 2017 12 / 24
Multiplicative Diophantine approximation Littlewood’s conjecture (LC), (1930): For any α 1 , α 2 ∈ R and any ε > 0 there exists q ∈ Z � =0 , p 1 , p 2 ∈ Z such that | q | · | q α 1 − p 1 | · | q α 2 − p 2 | < ε . What’s known: • LC holds if either α 1 or α 2 is not badly approximable. • LC holds if α 1 and α 2 lie in the same cubic field (Cassels & Swinnerton-Dyer, 1955) • V. Beresnevich (University of York) Diophantine approximation 18 January 2017 12 / 24
Multiplicative Diophantine approximation Littlewood’s conjecture (LC), (1930): For any α 1 , α 2 ∈ R and any ε > 0 there exists q ∈ Z � =0 , p 1 , p 2 ∈ Z such that | q | · | q α 1 − p 1 | · | q α 2 − p 2 | < ε . What’s known: • LC holds if either α 1 or α 2 is not badly approximable. • LC holds if α 1 and α 2 lie in the same cubic field (Cassels & Swinnerton-Dyer, 1955) • The set of counterexample to LC has zero dimension (Einsiedler, Katok and Lindenstrauss, 2006), led Lindenstrauss to win a Fields medal in 2010 • V. Beresnevich (University of York) Diophantine approximation 18 January 2017 12 / 24
Multiplicative Diophantine approximation Littlewood’s conjecture (LC), (1930): For any α 1 , α 2 ∈ R and any ε > 0 there exists q ∈ Z � =0 , p 1 , p 2 ∈ Z such that | q | · | q α 1 − p 1 | · | q α 2 − p 2 | < ε . What’s known: • LC holds if either α 1 or α 2 is not badly approximable. • LC holds if α 1 and α 2 lie in the same cubic field (Cassels & Swinnerton-Dyer, 1955) • The set of counterexample to LC has zero dimension (Einsiedler, Katok and Lindenstrauss, 2006), led Lindenstrauss to win a Fields medal in 2010 √ √ • It is not known if LC holds for ( 2 , 3) or any other explicit examples of badly approximable pairs. V. Beresnevich (University of York) Diophantine approximation 18 January 2017 12 / 24
Multiplicative Diophantine approximation Littlewood’s conjecture (LC), (1930): For any α 1 , α 2 ∈ R and any ε > 0 there exists q ∈ Z � =0 , p 1 , p 2 ∈ Z such that | q | · | q α 1 − p 1 | · | q α 2 − p 2 | < ε . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 13 / 24
Multiplicative Diophantine approximation Littlewood’s conjecture (LC), (1930): For any α 1 , α 2 ∈ R and any ε > 0 there exists q ∈ Z � =0 , p 1 , p 2 ∈ Z such that | q | · | q α 1 − p 1 | · | q α 2 − p 2 | < ε . Let 1 0 α 1 q α 1 − p Z 3 = { : q , p 1 , p 2 ∈ Z } . Λ = 0 1 α 2 q α 2 − p 0 0 1 q V. Beresnevich (University of York) Diophantine approximation 18 January 2017 13 / 24
Multiplicative Diophantine approximation Littlewood’s conjecture (LC), (1930): For any α 1 , α 2 ∈ R and any ε > 0 there exists q ∈ Z � =0 , p 1 , p 2 ∈ Z such that | q | · | q α 1 − p 1 | · | q α 2 − p 2 | < ε . Let 1 0 α 1 q α 1 − p Z 3 = { : q , p 1 , p 2 ∈ Z } . Λ = 0 1 α 2 q α 2 − p 0 0 1 q LC: ( λ 1 ,λ 2 ,λ 3 ) ∈ Λ , λ 3 � =0 | λ 1 λ 2 λ 3 | = 0 . inf V. Beresnevich (University of York) Diophantine approximation 18 January 2017 13 / 24
Multiplicative Diophantine approximation Littlewood’s conjecture (LC), (1930): For any α 1 , α 2 ∈ R and any ε > 0 there exists q ∈ Z � =0 , p 1 , p 2 ∈ Z such that | q | · | q α 1 − p 1 | · | q α 2 − p 2 | < ε . Let 1 0 α 1 q α 1 − p Z 3 = { : q , p 1 , p 2 ∈ Z } . Λ = 0 1 α 2 q α 2 − p 0 0 1 q LC: ( λ 1 ,λ 2 ,λ 3 ) ∈ Λ , λ 3 � =0 | λ 1 λ 2 λ 3 | = 0 . inf Admissible lattices: Λ ∈ L n := GL ( n , R ) / SL ( n , Z ) is admissible if Nm (Λ) := inf {| λ 1 · · · λ n | : ( λ 1 , . . . , λ n ) ∈ Λ � = 0 } > 0 . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 13 / 24
Multiplicative Diophantine approximation Littlewood’s conjecture (LC), (1930): For any α 1 , α 2 ∈ R and any ε > 0 there exists q ∈ Z � =0 , p 1 , p 2 ∈ Z such that | q | · | q α 1 − p 1 | · | q α 2 − p 2 | < ε . Let 1 0 α 1 q α 1 − p Z 3 = { : q , p 1 , p 2 ∈ Z } . Λ = 0 1 α 2 q α 2 − p 0 0 1 q LC: ( λ 1 ,λ 2 ,λ 3 ) ∈ Λ , λ 3 � =0 | λ 1 λ 2 λ 3 | = 0 . inf Admissible lattices: Λ ∈ L n := GL ( n , R ) / SL ( n , Z ) is admissible if Nm (Λ) := inf {| λ 1 · · · λ n | : ( λ 1 , . . . , λ n ) ∈ Λ � = 0 } > 0 . Fact: The set of admissible lattices is dense in L n . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 13 / 24
Multiplicative Diophantine approximation Littlewood’s conjecture (LC), (1930): For any α 1 , α 2 ∈ R and any ε > 0 there exists q ∈ Z � =0 , p 1 , p 2 ∈ Z such that | q | · | q α 1 − p 1 | · | q α 2 − p 2 | < ε . Let 1 0 α 1 q α 1 − p Z 3 = { : q , p 1 , p 2 ∈ Z } . Λ = 0 1 α 2 q α 2 − p 0 0 1 q LC: ( λ 1 ,λ 2 ,λ 3 ) ∈ Λ , λ 3 � =0 | λ 1 λ 2 λ 3 | = 0 . inf Admissible lattices: Λ ∈ L n := GL ( n , R ) / SL ( n , Z ) is admissible if Nm (Λ) := inf {| λ 1 · · · λ n | : ( λ 1 , . . . , λ n ) ∈ Λ � = 0 } > 0 . Fact: The set of admissible lattices is dense in L n . n = 2: V. Beresnevich (University of York) Diophantine approximation 18 January 2017 13 / 24
Multiplicative Diophantine approximation Littlewood’s conjecture (LC), (1930): For any α 1 , α 2 ∈ R and any ε > 0 there exists q ∈ Z � =0 , p 1 , p 2 ∈ Z such that | q | · | q α 1 − p 1 | · | q α 2 − p 2 | < ε . Let 1 0 α 1 q α 1 − p Z 3 = { : q , p 1 , p 2 ∈ Z } . Λ = 0 1 α 2 q α 2 − p 0 0 1 q LC: ( λ 1 ,λ 2 ,λ 3 ) ∈ Λ , λ 3 � =0 | λ 1 λ 2 λ 3 | = 0 . inf Admissible lattices: Λ ∈ L n := GL ( n , R ) / SL ( n , Z ) is admissible if Nm (Λ) := inf {| λ 1 · · · λ n | : ( λ 1 , . . . , λ n ) ∈ Λ � = 0 } > 0 . Fact: The set of admissible lattices is dense in L n . n = 2: � a � b Z 2 is admissible Λ = c d V. Beresnevich (University of York) Diophantine approximation 18 January 2017 13 / 24
Multiplicative Diophantine approximation Littlewood’s conjecture (LC), (1930): For any α 1 , α 2 ∈ R and any ε > 0 there exists q ∈ Z � =0 , p 1 , p 2 ∈ Z such that | q | · | q α 1 − p 1 | · | q α 2 − p 2 | < ε . Let 1 0 α 1 q α 1 − p Z 3 = { : q , p 1 , p 2 ∈ Z } . Λ = 0 1 α 2 q α 2 − p 0 0 1 q LC: ( λ 1 ,λ 2 ,λ 3 ) ∈ Λ , λ 3 � =0 | λ 1 λ 2 λ 3 | = 0 . inf Admissible lattices: Λ ∈ L n := GL ( n , R ) / SL ( n , Z ) is admissible if Nm (Λ) := inf {| λ 1 · · · λ n | : ( λ 1 , . . . , λ n ) ∈ Λ � = 0 } > 0 . Fact: The set of admissible lattices is dense in L n . n = 2: � a � b Z 2 is admissible ⇐ Λ = ⇒ c d V. Beresnevich (University of York) Diophantine approximation 18 January 2017 13 / 24
Multiplicative Diophantine approximation Littlewood’s conjecture (LC), (1930): For any α 1 , α 2 ∈ R and any ε > 0 there exists q ∈ Z � =0 , p 1 , p 2 ∈ Z such that | q | · | q α 1 − p 1 | · | q α 2 − p 2 | < ε . Let 1 0 α 1 q α 1 − p Z 3 = { : q , p 1 , p 2 ∈ Z } . Λ = 0 1 α 2 q α 2 − p 0 0 1 q LC: ( λ 1 ,λ 2 ,λ 3 ) ∈ Λ , λ 3 � =0 | λ 1 λ 2 λ 3 | = 0 . inf Admissible lattices: Λ ∈ L n := GL ( n , R ) / SL ( n , Z ) is admissible if Nm (Λ) := inf {| λ 1 · · · λ n | : ( λ 1 , . . . , λ n ) ∈ Λ � = 0 } > 0 . Fact: The set of admissible lattices is dense in L n . n = 2: � a � ⇒ a b , c b Z 2 is admissible ⇐ Λ = d ∈ Bad = Bad (1 , 1) ⊂ R . c d V. Beresnevich (University of York) Diophantine approximation 18 January 2017 13 / 24
Admissible lattices: examples Admissible lattices: Λ ∈ L n := GL ( n , R ) / SL ( n , Z ) is admissible if Nm (Λ) := inf {| λ 1 · · · λ n | : ( λ 1 , . . . , λ n ) ∈ Λ � = 0 } > 0 . n = 2: � a � ⇒ a b , c b Z 2 is admissible ⇐ Λ = d ∈ Bad = Bad (1 , 1) ⊂ R . c d Example (admissible lattices in L n for n ≥ 3): V. Beresnevich (University of York) Diophantine approximation 18 January 2017 14 / 24
Admissible lattices: examples Admissible lattices: Λ ∈ L n := GL ( n , R ) / SL ( n , Z ) is admissible if Nm (Λ) := inf {| λ 1 · · · λ n | : ( λ 1 , . . . , λ n ) ∈ Λ � = 0 } > 0 . n = 2: � a � ⇒ a b , c b Z 2 is admissible ⇐ Λ = d ∈ Bad = Bad (1 , 1) ⊂ R . c d Example (admissible lattices in L n for n ≥ 3): Let f ∈ Z [ x ], monic, irreducible over Q , deg f = n with all real roots. Thus f ( x ) = ( x − α 1 ) . . . ( x − α n ) with α i ∈ R . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 14 / 24
Admissible lattices: examples Admissible lattices: Λ ∈ L n := GL ( n , R ) / SL ( n , Z ) is admissible if Nm (Λ) := inf {| λ 1 · · · λ n | : ( λ 1 , . . . , λ n ) ∈ Λ � = 0 } > 0 . n = 2: � a � ⇒ a b , c b Z 2 is admissible ⇐ Λ = d ∈ Bad = Bad (1 , 1) ⊂ R . c d Example (admissible lattices in L n for n ≥ 3): Let f ∈ Z [ x ], monic, irreducible over Q , deg f = n with all real roots. Thus f ( x ) = ( x − α 1 ) . . . ( x − α n ) with α i ∈ R . Define α n − 1 1 α 1 . . . 1 . . ... Z n . . . Λ = . . α n − 1 1 α n . . . n V. Beresnevich (University of York) Diophantine approximation 18 January 2017 14 / 24
Admissible lattices: examples Admissible lattices: Λ ∈ L n := GL ( n , R ) / SL ( n , Z ) is admissible if Nm (Λ) := inf {| λ 1 · · · λ n | : ( λ 1 , . . . , λ n ) ∈ Λ � = 0 } > 0 . n = 2: � a � ⇒ a b , c b Z 2 is admissible ⇐ Λ = d ∈ Bad = Bad (1 , 1) ⊂ R . c d Example (admissible lattices in L n for n ≥ 3): Let f ∈ Z [ x ], monic, irreducible over Q , deg f = n with all real roots. Thus f ( x ) = ( x − α 1 ) . . . ( x − α n ) with α i ∈ R . Define α n − 1 1 α 1 . . . 1 . . ... Z n . . . Λ = . . α n − 1 1 α n . . . n � � � Nm (Λ) = inf | g ( α i ) | : g ∈ Z [ x ] � =0 , deg g ≤ n − 1 1 ≤ i ≤ n � � = inf | Resultant ( f , g ) | : g ∈ Z [ x ] � =0 , deg g ≤ n − 1 ≥ 1 . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 14 / 24
Admissible lattices and Bad V. Beresnevich (University of York) Diophantine approximation 18 January 2017 15 / 24
Admissible lattices and Bad Let g be an ( n + 1) × ( n + 1) matrix such that Λ = g Z n +1 is admissible. Let ( x 0 , x 1 , . . . , x n ) be any row (column) of g . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 15 / 24
Admissible lattices and Bad Let g be an ( n + 1) × ( n + 1) matrix such that Λ = g Z n +1 is admissible. Let ( x 0 , x 1 , . . . , x n ) be any row (column) of g . Then x 0 � = 0 and � x 1 � , . . . , x n ∈ Bad ( n , 1) . x 0 x 0 Proof. V. Beresnevich (University of York) Diophantine approximation 18 January 2017 15 / 24
Admissible lattices and Bad Let g be an ( n + 1) × ( n + 1) matrix such that Λ = g Z n +1 is admissible. Let ( x 0 , x 1 , . . . , x n ) be any row (column) of g . Then x 0 � = 0 and � x 1 � , . . . , x n ∈ Bad ( n , 1) . x 0 x 0 Proof. WLOG x 0 = 1 and g is the first column. V. Beresnevich (University of York) Diophantine approximation 18 January 2017 15 / 24
Admissible lattices and Bad Let g be an ( n + 1) × ( n + 1) matrix such that Λ = g Z n +1 is admissible. Let ( x 0 , x 1 , . . . , x n ) be any row (column) of g . Then x 0 � = 0 and � x 1 � , . . . , x n ∈ Bad ( n , 1) . x 0 x 0 Proof. WLOG x 0 = 1 and g is the first column. Let Q > 1 and ( q 1 , . . . , q n , p ) ∈ Z n +1 � = 0 with | q i | ≤ Q . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 15 / 24
Admissible lattices and Bad Let g be an ( n + 1) × ( n + 1) matrix such that Λ = g Z n +1 is admissible. Let ( x 0 , x 1 , . . . , x n ) be any row (column) of g . Then x 0 � = 0 and � x 1 � , . . . , x n ∈ Bad ( n , 1) . x 0 x 0 Proof. WLOG x 0 = 1 and g is the first column. Let Q > 1 and ( q 1 , . . . , q n , p ) ∈ Z n +1 � = 0 with | q i | ≤ Q . Recall, 0 < Nm (Λ) ≤ | λ 0 · · · λ n | , ( λ 1 , . . . , λ n ) ∈ Λ � = 0 . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 15 / 24
Admissible lattices and Bad Let g be an ( n + 1) × ( n + 1) matrix such that Λ = g Z n +1 is admissible. Let ( x 0 , x 1 , . . . , x n ) be any row (column) of g . Then x 0 � = 0 and � x 1 � , . . . , x n ∈ Bad ( n , 1) . x 0 x 0 Proof. WLOG x 0 = 1 and g is the first column. Let Q > 1 and ( q 1 , . . . , q n , p ) ∈ Z n +1 � = 0 with | q i | ≤ Q . Recall, 0 < Nm (Λ) ≤ | λ 0 · · · λ n | , ( λ 1 , . . . , λ n ) ∈ Λ � = 0 . Let ( λ 0 , . . . , λ n ) T = g ( − p , q 1 , . . . , q n ) T � = 0 . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 15 / 24
Admissible lattices and Bad Let g be an ( n + 1) × ( n + 1) matrix such that Λ = g Z n +1 is admissible. Let ( x 0 , x 1 , . . . , x n ) be any row (column) of g . Then x 0 � = 0 and � x 1 � , . . . , x n ∈ Bad ( n , 1) . x 0 x 0 Proof. WLOG x 0 = 1 and g is the first column. Let Q > 1 and ( q 1 , . . . , q n , p ) ∈ Z n +1 � = 0 with | q i | ≤ Q . Recall, 0 < Nm (Λ) ≤ | λ 0 · · · λ n | , ( λ 1 , . . . , λ n ) ∈ Λ � = 0 . Let ( λ 0 , . . . , λ n ) T = g ( − p , q 1 , . . . , q n ) T � = 0 . Note | λ i | ≤ C g Q . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 15 / 24
Admissible lattices and Bad Let g be an ( n + 1) × ( n + 1) matrix such that Λ = g Z n +1 is admissible. Let ( x 0 , x 1 , . . . , x n ) be any row (column) of g . Then x 0 � = 0 and � x 1 � , . . . , x n ∈ Bad ( n , 1) . x 0 x 0 Proof. WLOG x 0 = 1 and g is the first column. Let Q > 1 and ( q 1 , . . . , q n , p ) ∈ Z n +1 � = 0 with | q i | ≤ Q . Recall, 0 < Nm (Λ) ≤ | λ 0 · · · λ n | , ( λ 1 , . . . , λ n ) ∈ Λ � = 0 . Let ( λ 0 , . . . , λ n ) T = g ( − p , q 1 , . . . , q n ) T � = 0 . Note | λ i | ≤ C g Q . Then | λ 0 | = V. Beresnevich (University of York) Diophantine approximation 18 January 2017 15 / 24
Admissible lattices and Bad Let g be an ( n + 1) × ( n + 1) matrix such that Λ = g Z n +1 is admissible. Let ( x 0 , x 1 , . . . , x n ) be any row (column) of g . Then x 0 � = 0 and � x 1 � , . . . , x n ∈ Bad ( n , 1) . x 0 x 0 Proof. WLOG x 0 = 1 and g is the first column. Let Q > 1 and ( q 1 , . . . , q n , p ) ∈ Z n +1 � = 0 with | q i | ≤ Q . Recall, 0 < Nm (Λ) ≤ | λ 0 · · · λ n | , ( λ 1 , . . . , λ n ) ∈ Λ � = 0 . Let ( λ 0 , . . . , λ n ) T = g ( − p , q 1 , . . . , q n ) T � = 0 . Note | λ i | ≤ C g Q . Then | λ 0 | = | q 1 x 1 + · · · + q n x n − p | ≥ V. Beresnevich (University of York) Diophantine approximation 18 January 2017 15 / 24
Admissible lattices and Bad Let g be an ( n + 1) × ( n + 1) matrix such that Λ = g Z n +1 is admissible. Let ( x 0 , x 1 , . . . , x n ) be any row (column) of g . Then x 0 � = 0 and � x 1 � , . . . , x n ∈ Bad ( n , 1) . x 0 x 0 Proof. WLOG x 0 = 1 and g is the first column. Let Q > 1 and ( q 1 , . . . , q n , p ) ∈ Z n +1 � = 0 with | q i | ≤ Q . Recall, 0 < Nm (Λ) ≤ | λ 0 · · · λ n | , ( λ 1 , . . . , λ n ) ∈ Λ � = 0 . Let ( λ 0 , . . . , λ n ) T = g ( − p , q 1 , . . . , q n ) T � = 0 . Note | λ i | ≤ C g Q . Then | λ 0 | = | q 1 x 1 + · · · + q n x n − p | ≥ | Nm (Λ) | | λ 1 · · · λ n | ≥ V. Beresnevich (University of York) Diophantine approximation 18 January 2017 15 / 24
Admissible lattices and Bad Let g be an ( n + 1) × ( n + 1) matrix such that Λ = g Z n +1 is admissible. Let ( x 0 , x 1 , . . . , x n ) be any row (column) of g . Then x 0 � = 0 and � x 1 � , . . . , x n ∈ Bad ( n , 1) . x 0 x 0 Proof. WLOG x 0 = 1 and g is the first column. Let Q > 1 and ( q 1 , . . . , q n , p ) ∈ Z n +1 � = 0 with | q i | ≤ Q . Recall, 0 < Nm (Λ) ≤ | λ 0 · · · λ n | , ( λ 1 , . . . , λ n ) ∈ Λ � = 0 . Let ( λ 0 , . . . , λ n ) T = g ( − p , q 1 , . . . , q n ) T � = 0 . Note | λ i | ≤ C g Q . Then | λ 0 | = | q 1 x 1 + · · · + q n x n − p | ≥ | Nm (Λ) | | λ 1 · · · λ n | ≥ | Nm (Λ) | . C n g Q n V. Beresnevich (University of York) Diophantine approximation 18 January 2017 15 / 24
Admissible lattices and Bad Let g be an ( n + 1) × ( n + 1) matrix such that Λ = g Z n +1 is admissible. Let ( x 0 , x 1 , . . . , x n ) be any row (column) of g . Then x 0 � = 0 and � x 1 � , . . . , x n ∈ Bad ( n , 1) . x 0 x 0 Proof. WLOG x 0 = 1 and g is the first column. Let Q > 1 and ( q 1 , . . . , q n , p ) ∈ Z n +1 � = 0 with | q i | ≤ Q . Recall, 0 < Nm (Λ) ≤ | λ 0 · · · λ n | , ( λ 1 , . . . , λ n ) ∈ Λ � = 0 . Let ( λ 0 , . . . , λ n ) T = g ( − p , q 1 , . . . , q n ) T � = 0 . Note | λ i | ≤ C g Q . Then | λ 0 | = | q 1 x 1 + · · · + q n x n − p | ≥ | Nm (Λ) | | λ 1 · · · λ n | ≥ | Nm (Λ) | . C n g Q n Similarly one can give examples of matrices in Bad ( n , m ). V. Beresnevich (University of York) Diophantine approximation 18 January 2017 15 / 24
Admissible lattices and Bad Let g be an ( n + 1) × ( n + 1) matrix such that Λ = g Z n +1 is admissible. Let ( x 0 , x 1 , . . . , x n ) be any row (column) of g . Then x 0 � = 0 and � x 1 � , . . . , x n ∈ Bad ( n , 1) . x 0 x 0 Proof. WLOG x 0 = 1 and g is the first column. Let Q > 1 and ( q 1 , . . . , q n , p ) ∈ Z n +1 � = 0 with | q i | ≤ Q . Recall, 0 < Nm (Λ) ≤ | λ 0 · · · λ n | , ( λ 1 , . . . , λ n ) ∈ Λ � = 0 . Let ( λ 0 , . . . , λ n ) T = g ( − p , q 1 , . . . , q n ) T � = 0 . Note | λ i | ≤ C g Q . Then | λ 0 | = | q 1 x 1 + · · · + q n x n − p | ≥ | Nm (Λ) | | λ 1 · · · λ n | ≥ | Nm (Λ) | . C n g Q n Similarly one can give examples of matrices in Bad ( n , m ). The proof does not extend to weighted badly approximable points. V. Beresnevich (University of York) Diophantine approximation 18 January 2017 15 / 24
More on Bad Q: How likely is that a random matrix lies in Bad ( n , m )? Answer: V. Beresnevich (University of York) Diophantine approximation 18 January 2017 16 / 24
More on Bad Q: How likely is that a random matrix lies in Bad ( n , m )? Answer: Unlikely! The set Bad ( n , m ) has Lebesgue measure 0. V. Beresnevich (University of York) Diophantine approximation 18 January 2017 16 / 24
More on Bad Q: How likely is that a random matrix lies in Bad ( n , m )? Answer: Unlikely! The set Bad ( n , m ) has Lebesgue measure 0. However: Bad ( n , m ) has Hausdorff dimension nm , the same as the dimension of the ambient space. (Schmidt, 1960s) V. Beresnevich (University of York) Diophantine approximation 18 January 2017 16 / 24
More on Bad Q: How likely is that a random matrix lies in Bad ( n , m )? Answer: Unlikely! The set Bad ( n , m ) has Lebesgue measure 0. However: Bad ( n , m ) has Hausdorff dimension nm , the same as the dimension of the ambient space. (Schmidt, 1960s) Bad := Bad (1 , 1) can be characterise using continued fractions: V. Beresnevich (University of York) Diophantine approximation 18 January 2017 16 / 24
More on Bad Q: How likely is that a random matrix lies in Bad ( n , m )? Answer: Unlikely! The set Bad ( n , m ) has Lebesgue measure 0. However: Bad ( n , m ) has Hausdorff dimension nm , the same as the dimension of the ambient space. (Schmidt, 1960s) Bad := Bad (1 , 1) can be characterise using continued fractions: 1 x = a 0 + = [ a 0 ; a 1 , a 2 , . . . ] , 1 a 1 + a 2 + ... where all a n ∈ Z , a n ≥ 1 for n = 1 , 2 , . . . . V. Beresnevich (University of York) Diophantine approximation 18 January 2017 16 / 24
More on Bad Q: How likely is that a random matrix lies in Bad ( n , m )? Answer: Unlikely! The set Bad ( n , m ) has Lebesgue measure 0. However: Bad ( n , m ) has Hausdorff dimension nm , the same as the dimension of the ambient space. (Schmidt, 1960s) Bad := Bad (1 , 1) can be characterise using continued fractions: 1 x = a 0 + = [ a 0 ; a 1 , a 2 , . . . ] , 1 a 1 + a 2 + ... where all a n ∈ Z , a n ≥ 1 for n = 1 , 2 , . . . . The convergents p n := [ a 1 , a 2 , a 3 , . . . , a n ] q n are best approximations and satisfy 1 � � 1 � x − p n � � < � < . ( a n +1 + 2) q 2 a n +1 q 2 q n n n V. Beresnevich (University of York) Diophantine approximation 18 January 2017 16 / 24
More on Bad Q: How likely is that a random matrix lies in Bad ( n , m )? Answer: Unlikely! The set Bad ( n , m ) has Lebesgue measure 0. However: Bad ( n , m ) has Hausdorff dimension nm , the same as the dimension of the ambient space. (Schmidt, 1960s) Bad := Bad (1 , 1) can be characterise using continued fractions: 1 x = a 0 + = [ a 0 ; a 1 , a 2 , . . . ] , 1 a 1 + a 2 + ... where all a n ∈ Z , a n ≥ 1 for n = 1 , 2 , . . . . The convergents p n := [ a 1 , a 2 , a 3 , . . . , a n ] q n are best approximations and satisfy 1 � � 1 � x − p n � � < � < . ( a n +1 + 2) q 2 a n +1 q 2 q n n n Fact: An irrational x = [ a 0 ; a 1 , a 2 . . . ] is in Bad ⇔ ∃ M ∀ i ∈ N a i ≤ M V. Beresnevich (University of York) Diophantine approximation 18 January 2017 16 / 24
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