Uniform distribution of prime powers and sets of recurrence and van der Corput sets in ℤ k

Bergelson, Vitaly; Kolesnik, Grigori; Madritsch, Manfred G.; Son, Y. G.; Tichy, Robert F.

doi:10.1007/s11856-014-1049-4

Cited by 16 publications

(16 citation statements)

References 22 publications

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“…Each of these examples lead to different exponential sums whose treatment is interesting on their own. In the vain of Bergelson et al [7], where mixtures of polynomials and pseudo-polynomials were considered, similar results should hold for Heilbronn sets. For example, let f be a polynomial with real coefficients.…”

Section: Final Remarksmentioning

confidence: 62%

“…Note that in the case of 0 < |β| < N ρ−deg f we apply a differnt argument that allows us to reuse the estimates for bigger β. Furthermore we note that the exponent 1 10 is an artifact of Lemma 2.3 of [7] which we use in the proof. If θ r > k we may apply Weyl-differencing sufficiently often till the sum does not rotate to much.…”

Section: Exponential Sum Estimates For the Case θ R > Kmentioning

confidence: 96%

“…[7, Lemma 2.3]). Assume F (x) to be any function defined on the real line, supported on [X, 2X] and bounded by F 0 .…”

mentioning

confidence: 99%

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Multidimensional Van Der Corput Sets and Small Fractional Parts of Polynomials

Madritsch

Tichy

2019

Mathematika

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We establish Diophantine inequalities for the fractional parts of generalized polynomials f , in particular for sequences ν(n) = ⌊n c ⌋ + n k with c > 1 a non-integral real number and k ∈ N, as well as for ν(p) where p runs through all prime numbers. This is related to classical work of Heilbronn and to recent results of Bergelson et al.considered the Diophantine inequality over smooth numbers to obtain an improvement. The proofs of these results are based on a sophisticated treatment of the occurring exponential sums. In a recent paper Lê and Spencer [29] proved the following Theorem 1.2 ([29, Theorem 3]). Let N ∈ N and h ∈ Z[X] be a polynomial with integer coefficients such that for every non-zero integer q there exists a solution n to the congruence h(n) ≡ 0 mod q. Then there is an exponent η > 0 depending only on the degree of h such that min 1≤n≤N ξh(n) ≪ h N −η for arbitrary ξ ∈ R.

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Section: Final Remarksmentioning

confidence: 62%

Section: Exponential Sum Estimates For the Case θ R > Kmentioning

confidence: 96%

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Multidimensional Van Der Corput Sets and Small Fractional Parts of Polynomials

Madritsch

Tichy

2019

Mathematika

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“…Prime powers. In a recent paper the authors together with Bergelson, Kolesnik and Son [3] consider sets of the form…”

Section: Various Examples and Applications To Additive Problemsmentioning

confidence: 99%

Dynamical Systems and Uniform Distribution of Sequences

Madritsch

Tichy

2016

From Arithmetic to Zeta-Functions

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“…where the supremum is over all blocks of length ℓ. Then a number θ is normal to base q if for each ℓ ≥ 1 we have that R N,ℓ (θ) = o (1) for N → ∞. Furthermore we call a number absolutely normal if it is normal in all bases q ≥ 2.…”

Section: Introductionmentioning

confidence: 99%