2019
DOI: 10.1112/s0025579318000529
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Multidimensional Van Der Corput Sets and Small Fractional Parts of Polynomials

Abstract: 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 exponent… Show more

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Cited by 7 publications
(10 citation statements)
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“…Goal of the present note. In an earlier work (see [13]) we improved the results in [12] for the case of dominant f . More specifically, we obtained Theorem 1.2.…”
Section: Introduction and Statement Of Resultsmentioning
confidence: 81%
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“…Goal of the present note. In an earlier work (see [13]) we improved the results in [12] for the case of dominant f . More specifically, we obtained Theorem 1.2.…”
Section: Introduction and Statement Of Resultsmentioning
confidence: 81%
“…For a polynomial with property (F), our result improves on the exponent obtained by Madritsch and Tichy. In particular, the exponent (1.6) is lower bounded by the reciprocal of quartic polynomial in the degree of f , while the exponent obtained in [12] decays exponentially in the degree of f .…”
Section: Statement Of Resultsmentioning
confidence: 97%
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“…These and related questions have been the object of a large amount of study in analytic number theory; see [12,6,2,1,3,18,22,20,7,15,23] for some recent related work. We refer the reader to the book [4] for a comprehensive overview of these questions.…”
Section: Introductionmentioning
confidence: 99%