Bohr sets in sumsets I: Compact groups

Le, Anh N.; Lê, Thái Hoàng

doi:10.48550/arxiv.2112.11997

Cited by 3 publications

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“…As a consequence of Theorems 1.2 and 1.4, we obtain immediately the following number field generalization of Theorems B and C. In [32], this result was proved (at least for Z[i]) using a different argument, similar to Bogolyubov and Bergelson-Ruzsa's proofs of Theorems A and B in Z.…”

Section: Introductionmentioning

confidence: 60%

“…In [32], these objectives were achieved for compact abelian groups. Note that in this case, the only invariant mean on G is given by m G (the normalized Haar measure on G) and d * (A) = m G (A).…”

Section: Introductionmentioning

confidence: 99%

“…In order to utilize the corresponding result in compact groups [32], we need to show that the homomorphisms φ 1 , φ 2 , φ 3 induce homomorphisms φj on Z satisfying φj •τ = τ •φ j . This is straightforward under the additional assumption that spectrum of (X, µ, T ) (i.e.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, any extension of Z in Y will also be characteristic for these averages. The group rotation factor K of Y corresponding to Γ is such an extension of Z, and this allows us to transfer the Bohr sets obtained in [32] to G. The diagram below demonstrates the relations among X, Y, Z and K where…”

Section: Introductionmentioning

confidence: 99%

“…This investigation leads to the introduction of Radon-Nikodym densities ρ ν A , ρ ν φ 2 (A) and their relationship in Section 4. After the required relationship is established, we put all ingredients together (Proposition 7.1, Corollary 4.10) and use the compact counterpart in [32] to prove Theorem 1.4.…”

Section: Introductionmentioning

confidence: 99%

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Bohr sets in sumsets II: countable abelian groups

Griesmer¹,

Le²,

Lê³

2022

Preprint

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We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, letting G be a countable discrete abelian group and φ1, φ2, φ3 : G → G be commuting endomorphisms whose images have finite indices, we show that (1) If A ⊂ G has positive upper Banach density and φ1 + φ2 + φ3 = 0, then φ1(A) + φ2(A) + φ3(A) contains a Bohr set. This generalizes a theorem of Bergelson and Ruzsa in Z and a recent result of the first author.(2) For any partition G = r i=1 Ai, there exists an i ∈ {1, . . . , r} such that φ1(Ai) + φ2(Ai) − φ2(Ai) contains a Bohr set. This generalizes a result of the second and third authors from Z to countable abelian groups.(3) If B, C ⊂ G have positive upper Banach density and G = r i=1 Ai is a partition, B + C + Ai contains a Bohr set for some i ∈ {1, . . . , r}. This is a strengthening of a theorem of Bergelson, Furstenberg, and Weiss. All results are quantitative in the sense that the radius and rank of the Bohr set obtained depends only on the indices [G : φj(G)], the upper Banach density of A (in (1)), or the number of sets in the given partition (in (2) and ( 3)).

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Section: Introductionmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bohr sets in sumsets II: countable abelian groups

Griesmer¹,

Le²,

Lê³

2022

Preprint

Self Cite

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Bohr sets in sumsets II: countable abelian groups

Griesmer

Lê

2023

Forum of Mathematics, Sigma

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We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, letting G be a countable discrete abelian group and $\phi _1, \phi _2, \phi _3: G \to G$ be commuting endomorphisms whose images have finite indices, we show that (1) If $A \subset G$ has positive upper Banach density and $\phi _1 + \phi _2 + \phi _3 = 0$ , then $\phi _1(A) + \phi _2(A) + \phi _3(A)$ contains a Bohr set. This generalizes a theorem of Bergelson and Ruzsa in $\mathbb {Z}$ and a recent result of the first author. (2) For any partition $G = \bigcup _{i=1}^r A_i$ , there exists an $i \in \{1, \ldots , r\}$ such that $\phi _1(A_i) + \phi _2(A_i) - \phi _2(A_i)$ contains a Bohr set. This generalizes a result of the second and third authors from $\mathbb {Z}$ to countable abelian groups. (3) If $B, C \subset G$ have positive upper Banach density and $G = \bigcup _{i=1}^r A_i$ is a partition, $B + C + A_i$ contains a Bohr set for some $i \in \{1, \ldots , r\}$ . This is a strengthening of a theorem of Bergelson, Furstenberg and Weiss. All results are quantitative in the sense that the radius and rank of the Bohr set obtained depends only on the indices $[G:\phi _j(G)]$ , the upper Banach density of A (in (1)), or the number of sets in the given partition (in (2) and (3)).

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On Katznelson’s Question for skew-product systems

Glasscock

Koutsogiannis

Richter

2022

Bull. Amer. Math. Soc.

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Katznelson’s Question is a long-standing open question concerning recurrence in topological dynamics with strong historical and mathematical ties to open problems in combinatorics and harmonic analysis. In this article, we give a positive answer to Katznelson’s Question for certain towers of skew-product extensions of equicontinuous systems, including systems of the form ( x , t ) ↦ ( x + α , t + h ( x ) ) (x,t) \mapsto (x + \alpha , t + h(x)) . We describe which frequencies must be controlled for in order to ensure recurrence in such systems, and we derive combinatorial corollaries concerning the difference sets of syndetic subsets of the natural numbers.

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Bohr sets in sumsets I: Compact groups

Cited by 3 publications

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Bohr sets in sumsets II: countable abelian groups

Bohr sets in sumsets II: countable abelian groups

Bohr sets in sumsets II: countable abelian groups

On Katznelson’s Question for skew-product systems

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