New examples of complete sets, with connections to a Diophantine theorem of Furstenberg

Bergelson, Vitaly; Simmons, David

doi:10.4064/aa8221-10-2016

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A Direct Proof Of Corollaries 25 and 261

Introduction1

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2024

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Cited by 7 publications

(2 citation statements)

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“…Proof. -The proof is exactly the same as the preceding one, replacing the result from [16] by [9,Obs. 1.36].…”

Section: A Direct Proof Of Corollaries 25 and 26mentioning

confidence: 88%

Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences

Badea

Grivaux

2020

Comment. Math. Helv.

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“…Proof. -The proof is exactly the same as the preceding one, replacing the result from [16] by [9,Obs. 1.36].…”

Section: A Direct Proof Of Corollaries 25 and 26mentioning

confidence: 88%

Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences

Badea

Grivaux

2020

Comment. Math. Helv.

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“…Bergelson and Simmons [4] have recently considered sets related to {a m b n | m, n ∈ N}, where a, b ≥ 2 are again multiplicatively independent integers, but m and n are restricted to certain 'special' types of sets. We mention here some of their results.…”

Section: Introductionmentioning

confidence: 99%

A polynomial-exponential variation of Furstenberg’s theorem

Abramoff¹,

Berend²

2018

Ergod. Th. Dynam. Sys.

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Furstenberg’s $\times 2\times 3$ theorem asserts that the double sequence $(2^{m}3^{n}\unicode[STIX]{x1D6FC})_{m,n\geq 1}$ is dense modulo one for every irrational $\unicode[STIX]{x1D6FC}$. The same holds with $2$ and $3$ replaced by any two multiplicatively independent integers. Here we obtain the same result for the sequences $((\begin{smallmatrix}m+n\\ d\end{smallmatrix})a^{m}b^{n}\unicode[STIX]{x1D6FC})_{m,n\geq 1}$ for any non-negative integer $d$ and irrational $\unicode[STIX]{x1D6FC}$, and for the sequence $(P(m)a^{m}b^{n})_{m,n\geq 1}$, where $P$ is any polynomial with at least one irrational coefficient. Similarly to Furstenberg’s theorem, both results are obtained by considering appropriate dynamical systems.

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Density modulo 1 and Hardy fields

Abramoff,

Berend,

Kolesnik

2024

Isr. J. Math.

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In this article we extend Boshernitzan’s result on density modulo 1 of sequences arising from functions belonging to a Hardy field. We also merge these results with Furstenberg’s ×2×3 theorem. We prove, for example, that, given a vector f of subpolynomial functions in a Hardy field, such that $$(\mathbf{f}(n))_{n=1}^{\infty}$$ ( f ( n ) ) n = 1 ∞ is dense modulo 1 in Rd, the sequence (2m3nα, f(n))m,n≥1 is dense modulo 1 in Rd+1 for irrational α. Some negative results concerning Furstenberg’s theorem are obtained as well.

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New examples of complete sets, with connections to a Diophantine theorem of Furstenberg

Cited by 7 publications

References 12 publications

Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences

Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences

A polynomial-exponential variation of Furstenberg’s theorem

Density modulo 1 and Hardy fields

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