Improvements on the star discrepancy of (t,s)-sequences Henri Faure Institut de math´ ematiques de Luminy, Marseille, France faure@iml.univ-mrs.fr Joint work with Christiane Lemieux MCQMC2012, Sydney, February 13–17, 2012
Outline 1. Reminders on discrepancy and known bounds, p. 3–9 2. Applying Atanassov’s method to ( t, s )-sequences, p. 10–13 3. Improvement in the case of an even base, p. 14–16 4. Comparison of the constants, p. 17–19 5. New results on star discrepancy of ( t, 1)-sequences, p. 20–22 2
Discrepancy For a point set P N = { X 1 , X 2 , ..., X N } in I s = [0 , 1) s and a subinterval J of I s , we define the discrepancy function as E ( J ; P N ) = A ( J ; P N ) − NV ( J ) where A ( J ; P N ) = # { n ; 1 ≤ n ≤ N, X n ∈ J } and V ( J ) is the volume of J . Then, the star (extreme) discrepancy D ∗ , respectively the (extreme) discrepancy D of P N are defined as D ∗ ( P N ) = sup J ∗ | E ( J ∗ ; P N ) | and D ( P N ) = sup J | E ( J ; P N ) | where J ∗ (resp. J ) is of the form � s i =1 [0 , y i [ (resp. � s i =1 [ y i , z i [). 3
For an infinite sequence X , D ( N, X ) and D ∗ ( N, X ) denote the discrep- ancies of its first N points. Similarly, for the discrepancy function, we write E ( J ; N ; X ) = A ( J ; N ; X ) − NV ( J ). A sequence satisfying D ∗ ( N, X ) ∈ O ((log N ) s ) is typically considered to be a low-discrepancy sequence. But the constant hidden in the O notation needs to be made explicit to make comparisons possible across sequences. This is achieved with an inequality of the form D ∗ ( N, X ) ≤ c s (log N ) s + O ((log N ) s − 1 ) . (1) But this is still unsatisfactory and the constants hidden in the com- plementary term can be so huge that the leading term can lose any significance in applications. Nevertheless, improving the leading term c s is still of interest from a number theory point of view. 4
Low-discrepancy sequences Well known low-discrepancy sequences are Halton sequences and ( t, s )-sequences built with one-dimensional van der Corput sequences. A generalized van der Corput sequence S Σ in base b associated with b a sequence Σ = ( σ r ) r ≥ 0 of permutations of Z b = { 0 , 1 , . . . , b − 1 } is defined as ( n ≥ 1) � � ∞ ∞ a r ( n ) σ r S Σ a r ( n ) b r ( b-adic expansion) . � � b ( n ) = with n − 1 = b r +1 r =0 r =0 Original v.d.Corput sequences are obtained with σ r = id for all r ≥ 0. Generalized Halton sequences are obtained with S Σ sequences in co- b prime bases on each coordinate: ( S Σ 1 b 1 , . . . , S Σ s b s ). 5
The concept of (t,s)-sequences was introduced by Niederreiter to give a general framework for various constructions using generating ma- trices applied to v.d.Corput sequences (Sobol’, Faure, Niederreiter). – An elementary interval E in I s is defined as ( a i , d i are integers) s [ a i b − d i , ( a i + 1) b − d i ) with 0 ≤ a i ≤ b d i for 1 ≤ i ≤ s. � E = i =1 – Given integers t, m with 0 ≤ t ≤ m , a ( t, m, s ) − net in base b is an s -dimensional set with b m points such that any elementary interval in base b with volume b t − m contains exactly b t points of the set. – An s -dimensional sequence X in I s is a ( t, s )-sequence in base b if the subset { X ( n ) : kb m < n ≤ ( k + 1) b m } is a ( t, m, s ) − net in base b for all integers k ≥ 0 and m ≥ t. 6
However, to give sense to new important constructions, Tezuka and then Niederreiter and Xing in a general form needed a new definition. i =1 x i b − i be a b -adic expansion of x ∈ [0 , 1], Truncation : Let x = � ∞ with the possibility that x i = b − 1 for all but finitely many i . For every integer m ≥ 1, we define the m -truncation of x by i =1 x i b − i [ x ] b,m = � m If X ∈ I s , [ X ] b,m means an m -truncation is applied to each coordinate. An s -dimensional sequence ( X n ) n ≥ 1 , with prescribed b − adic expan- sions for each coordinate, is a ( t, s )-sequence (in the broad sense) if the subset { [ X n ] b,m ; kb m < n ≤ ( k + 1) b m } is a ( t, m, s )-net in base b for all integers k ≥ 0 and m ≥ t ≥ 0 . The former ( t, s )-sequences are now called ( t, s )-sequences in the nar- row sense and the others ( t, s )-sequences (except possible confusion). 7
Bounds for the discrepancy of ( t, s )-sequences Various constants c s below refer to inequality (1) for low-disc.seq. s ⌊ b = b t 2 ⌋ b − 1 c Ni For s ≥ 1 , (Niederreiter, 1987) . (2) s 2 ⌊ b s ! log b 2 ⌋ � s = b t b − 1 b � For s ≥ 2 , c Kr if b is an even base (3) s s ! 2( b + 1) 2 log b � b − 1 = b t � s 1 and c Kr if b is an odd base (Kritzer, 2006) , (4) s s ! 2 2 log b by a factor 1 b hence improving c Ni 2 for odd b and 2( b +1) for even b s in case of dimension s ≥ 2. For short, we leave out special cases. 8
Further Kritzer stated the conjecture that for s ≥ 2 and even b , � b − 1 b 2 = b t � s c conj (5) . s 2( b 2 − 1) s ! 2 log b As for ( t, s )-sequences in the broad sense, Niederreiter–Xing (1996) showed that constant c Ni in (2) is still valid, but Kritzer did not take s into account this generalization in his proofs. Our aim is to closely approach (5) for ( t, s )-sequences in the broad sense. To this end, we deepen the process from a preceding study which consists in using Atanassov’s method for Halton sequences to obtain discrepancy bounds for ( t, s )-sequences (proc. MCQMC 2010). If time permits, we will also provide a new result for the discrepancy of ( t, 1)-sequences for which the approach of Kritzer does not work . 9
Atanassov’s method applied to ( t, s )-seq. We have been able to adapt Atanassov’s proof for Halton sequences to a single base b and to ( t, s )-sequences in the broad sense, getting Theorem 1 For any ( t, s ) -sequence X (in the broad sense) we have � log N � log N s − 1 D ∗ ( N, X ) ≤ b t � s � k �� b b �� b + b t � log b + s log b + k . s ! 2 k ! 2 k =0 Corollary 1 The leading constant c s in (1) satisfies � b − 1 c s = b t � s if b is odd, and c s = b t � s b � if b is even. s ! 2 log b s ! 2 log b b For odd b , c s = c Ni but for even b , it is larger by a factor b − 1 . s 10
Idea of the proof � log N � Note P N ( X ) the set of the first N points of X and set n := . log b Let [ P N ( X )] := { ([ X (1) ] b,n +1 , . . . , [ X ( s ) ] b,n +1 ) , 1 ≤ k ≤ N } where [ . ] is k k the truncation operator and ( X (1) , . . . , X ( s ) ) is the k th term of X . k k We first prove Theorem 1 for this truncated version of X . (i) We use numeration systems in base b with signed digits: z ∈ [0 , 1) is written as ∞ with | a j | ≤ b − 1 if b is odd a j b − j � 2 z = with | a j | ≤ b 2 and | a j | + | a j +1 | ≤ b − 1 if b is even. j =0 (ii) We use signed splittings of an interval J ∈ I s , i.e any collection of intervals J 1 , . . . , J n and respective signs ǫ 1 , . . . , ǫ n ( ± 1), such that for n any additive function ν on intervals in I s , we have ν ( J ) = � ǫ i ν ( J i ) . i =1 11
(iii) We use signed splittings I ( j ) = I ( j 1 , . . . , j s ) deduced from (i) to get the discrepancy function: n n � � E ( J ; [ P N ( X )]) = · · · ǫ ( j ) E ( I ( j ); [ P N ( X )]) =: Σ 1 + Σ 2 j 1 =0 j s =0 where j ∈ Σ 1 ⇔ b j 1 . . . b j s ≤ N and Σ 2 is the complementary sum. (iv) Two main ingredients to bound Σ 1 and Σ 2 : i =1 [ b i b − d i , c i b − d i ) • | E ( J ; [ P N ( X )]) | ≤ b t � s i =1 ( c i − b i ) where J = � s (fundamental property of ( t, s )-sequences in the broad sense). � k ≤ 1 � log N � � j ; b j 1 . . . b j k ≤ N • # (from diophantine geometry). k ! log b Σ 1 gives the leading part of the bound ∈ (log N ) s and Σ 2 the other part ∈ (log N ) s − 1 in the statement of Theorem 1. 12
• This proves Theorem 1 for the truncated version of the sequence. • But, as shown by Niederreiter and ¨ Ozbudak in 2007 (Lemma 4.2, Acta Arith. 130, 79–97), when we go from the truncated to the untruncated version of the sequence, the bound for the discrepancy remains exactly the same. • Thus if a bound of the form (1) applies to the truncated version of a (t,s)-sequence, it applies to the untruncated version as well with the same constants. • In the case of an even base, a refinement of the method result- ing from a deeper investigation of Σ 1 allowed us to substantially of Kritzer, near to his conjecture c conj improve the bound c Kr : s s 13
Improvement in the case of an even base Theorem 2 For any ( t, s ) -sequence X (in the broad sense) in an even base b and for any N ≥ b s , we have � s � s − 1 D ∗ ( N, X ) ≤ b t � s � log N � ( b − 1) log N � b + sb t + s s ! 2 log b 2 log b s − 1 � k � b log N b + b t � 2 log b + k . k ! k =0 Sketch of the proof. At the outset, the proof is the same as for Theorem 1 until the discrepancy function E ( J ; [ P N ( X )]) is split up into Σ 1 and Σ 2 . The end of the proof, concerning Σ 2 , is also the same and gives the last term in the bound of Theorem 2. And the transition from the truncated to the untruncated version is the same too. Hence, what remains to be done is to deal with Σ 1 . 14
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