Groups and topologies related to D -sequences Daniel de la Barrera - PowerPoint PPT Presentation

Groups and topologies related to D -sequences Daniel de la Barrera Mayoral Universidad Complutense de Madrid Thanks: Ministerio de Econom´ ıa y Competitividad grant: MTM2013-42486-P . ORCID: 0000-0002-0024-5265. Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

Index Topologies on Z . 1 Topologies on Z ( b ∞ ) . 2 Topologies on Z b . 3 Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

p -adic topologies. Definition Let p be a prime number. The family p n Z : n ∈ N 0 � � is a neighborhood basis for a group topology on Z . This topology is called the p -adic topology, λ p . Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

p -adic topologies. Definition Let p be a prime number. The family p n Z : n ∈ N 0 � � is a neighborhood basis for a group topology on Z . This topology is called the p -adic topology, λ p . Properties λ p is metrizable. λ p is precompact. λ p is linear. λ p is locally quasi-convex. ( Z , λ p ) ∧ = Z ( p ∞ ) . Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

D -sequences. Definition Let b = ( b n ) n ∈ N 0 ⊂ N , satisfying that: b 0 = 1. b n | b n + 1 . b n � = b n + 1 . Then b is a D -sequence. Let D be the family of all D -sequences. For a D -sequence we define the sequence of b n ratios q n := b n − 1 . Example b n = p n is a D -sequence and q n = p for all n . b n = ( n + 1 )! is a D -sequence and q n = n + 1 for all n . Let q n be a sequence of natural numbers satisfying q n � = 1 for all n . Then b n := ∏ n i = 1 q i is a D -sequence. Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

D -sequences (2) Proposition Let b be a D-sequence. Suppose that q j + 1 � = 2 for infinitely many j. For each integer number L ∈ Z , there exists a natural number N = N ( L ) and unique integers k 0 ,..., k N , such that: N ∑ (1) L = k j b j . j = 0 � − q j + 1 2 , q j + 1 � (2) k j ∈ , for 0 ≤ j ≤ N. 2 Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

D -sequences (3) Proposition Let b be a D-sequence. Then any y ∈ R can be written uniquely in the form ∞ β n = β 0 + β 1 + β 2 + ··· + β s ∑ y = + ··· , ( 1 ) b n b 0 b 1 b 2 b s n = 0 where β n ∈ Z and | β n + 1 | ≤ q n + 1 2 . Further, n β j − 1 1 ∑ < y − ≤ 2 b n b j 2 b n j = 0 holds for all n ∈ N 0 . Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

D -sequences (4) Definition The sequence ∞ − q n 2 , q n � � ∏ ( k 0 , k 1 , ··· , k N ( L ) , 0 ,... ) ∈ , 2 n = 1 will be called the b-coordinates of L . The sequence ∞ − q n 2 , q n � � ∏ ( β 0 , β 1 ,... ) ∈ Z × 2 n = 1 will be called the b-coordinates of y . Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

D -sequences (5) D := { b : b is a D -sequence } D ∞ := { b ∈ D : b n + 1 b n → ∞ } . ∞ := { b ∈ D : b n + ℓ D ℓ b n → ∞ } . D ∞ ( b ) := { c ⊏ b : c ∈ D ∞ } . D ℓ ∞ ( b ) := { c ⊏ b : c ∈ D ℓ ∞ } . b has bounded ratios if there exists L such that q n < L for all n . b is basic if q n is a prime number for all n . � � Z ( b ∞ ) := � 1 b n + Z ≤ T . n ∈ N 0 − q n 2 , q n �� � � Z b := ∏ n ∈ N ∩ Z . 2 Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

b -adic topologies. Definition Let b be a D -sequence. The family { b n Z : n ∈ N 0 } is a neighborhood basis for a group topology on Z . This topology is called the b -adic topology, λ b . Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

b -adic topologies. Definition Let b be a D -sequence. The family { b n Z : n ∈ N 0 } is a neighborhood basis for a group topology on Z . This topology is called the b -adic topology, λ b . Properties λ b is metrizable. λ b is precompact. λ b is linear. λ b is locally quasi-convex. ( Z , λ b ) ∧ = Z ( b ∞ ) . Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

Topologies of uniform convergence on Z . In (Chasco et. al, 1999), the authors state that any locally quasi-convex group topology is the topology of uniform convergence on certain subsets of the dual. In order to define a topology of uniform convergence on Z we choose a family that is formed by only one subset which is precisely the range of a sequence b in T . Proposition Let b be a D-sequence. Fix b := { 1 b n + Z : n ∈ N 0 } ⊂ T . Define � � z ∈ Z : z V b , m := b n + Z ∈ T m for all n ∈ N 0 and V b := { V b , m : m ∈ N } . Then V b is a neighborhood basis for the topology of uniform convergence on b in the group of the integers. We call this topology τ b . Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

Basic properties of τ b . Properties Let b be a D-sequence. Then: τ b is a metrizable topology. χ ∈ b χ − 1 ( T m ) . Hence τ b is locally quasi-convex. V b , m = � Z ( b ∞ ) = � b � ≤ ( Z , τ b ) ∧ . Theorem Let b be a D-sequence such that b ∈ D ℓ ∞ . Then ( Z , τ b ) ∧ = Z ( b ∞ ) . Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

The topology δ b . Definition Let b be a D -sequence. Define δ b := sup { τ c : c ∈ D ℓ ∞ ( b ) } . Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

The topology δ b . Definition Let b be a D -sequence. Define δ b := sup { τ c : c ∈ D ℓ ∞ ( b ) } . Properties δ b is locally quasi-convex. ( Z , δ b ) ∧ = Z ( b ∞ ) . Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

δ b if b has bounded ratios. Theorem Let b be a basic D-sequence with bounded ratios and let λ b ( x n ) ⊂ Z be a non-quasiconstant sequence such that x n → 0 . Then there exists a metrizable locally quasi-convex compatible group topology τ (= τ c for some subsequence c of b ) on Z satisfying: ( a ) τ is compatible with λ b . τ ( b ) x n � 0 . ( c ) λ b < τ . Theorem Let b a basic D-sequence with bounded ratios. Then the topology δ b has no non-trivial convergent sequences. Hence, it cannot be non-discrete metrizable. Since ( Z , δ b ) ∧ � = T , it is non-discrete. Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

δ b if b has bounded ratios. (2) Remark If b is a basic D-sequence with bounded ratios, then D k ∞ ( b ) � = D k + 1 ( b ) . ∞ Remark If b is a basic D-sequence with bounded ratios, then τ b is discrete. Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

δ b if b ∈ D ℓ ∞ . Proposition Let b be a basic D-sequence such that b ∈ D ℓ ∞ . The following facts can be easily proved: The sequence b has unbounded ratios. We have that D k ∞ ( b ) = D ℓ ∞ ( b ) , for any natural number k ≥ ℓ . We have δ b = τ b . The topology δ b is metrizable. Hence, it has non-trivial convergent sequences. Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

δ b if b ∈ D ℓ ∞ . (2) Proposition Let b , c be the D-sequences defined as follows: • b 3 n + 1 = 2 · b 3 n • b 3 n + 2 = p n + 1 b 3 n + 1 • b 3 n + 3 = 3 · b 3 n + 2 and • c 3 n + 1 = 3 · c 3 n • c 3 n + 2 = p n + 1 c 3 n + 1 • c 3 n + 3 = 2 · c 3 n + 2 , where p n is the n-th prime number. Then Z ( b ∞ ) = Z ( c ∞ ) and δ b � = δ c . Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

The usual topology on Z ( b ∞ ) . Notation We shall denote by τ U the topology in Z ( b ∞ ) inherited from the one of the complex plane. Proposition (Aussenhofer, Chasco) Let H be a dense and metrizable subgroup of a topological group G. Then G ∧ = H ∧ . Corollary ( Z ( b ∞ ) , τ U ) ∧ is isomorphic to the discrete group of the integers. Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

The usual topology on Z ( b ∞ ) . Proposition β ( n ) Let ( x n + Z ) ⊂ Z ( b ∞ ) . Write x n = ∑ k . Then the following b k k ≥ 1 assertions are equivalent: x n + Z → 0 + Z in τ U . 1 | x n | → 0 in R . 2 For any k ∈ Z there exists n k such that β ( n ) = ··· = β ( n ) = 0 3 1 k if n ≥ n k . Daniel de la Barrera Mayoral Groups and topologies related to D -sequences

Hom ( Z ( b ∞ ) , T ) . We want to find a topology of uniform convergence on Z ( b ∞ ) . To that end we need a subset of Hom ( Z ( b ∞ ) , T ) ; i. e, of Z b , the group of b -adic integers. Describe the action of an element of Z b on Z ( b ∞ ) . Let k = ∑ k n b n ∈ Z b . We define χ k : Z ( b ∞ ) → Z where n ∈ N N ∑ χ ( x ) := xk + Z = lim k n b n x + Z . N → ∞ n = 0 Since x ∈ Z ( b ∞ ) implies that β n = 0 for n ≥ n 0 for some n 0 N ∑ we have that xb n ∈ Z if n ≥ n 0 and lim k n b n x + Z N → ∞ n = 0 stabilizes. Daniel de la Barrera Mayoral Groups and topologies related to D -sequences