Modular hyperbolas and Beatty sequences

Technau, Marc

doi:10.48550/arxiv.1808.00413

Cited by 2 publications

(3 citation statements)

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“…Here we study a question of distribution of modular inverses modulo a prime p of Piatetski-Shapiro sequences and, in fact, more general sequences. Our motivation comes from [29], where the distribution of inverses modulo a prime p of Beatty sequences is considered. Furthermore, since this question immediately leads us to a problem of estimating Kloosterman-like sums with Piatetski-Shapiro sequences, this serves as an additional motivation.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Kloosterman sums with twice-differentiable functions

Shparlinski¹,

Technau²

2020

Funct. Approx. Comment. Math.

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We bound Kloosterman-like sums of the shape N ÿ n"1 expp2πipxtf pnqu`ytf pnqu´1q{pq, with integers parts of a real-valued, twice-differentiable function f is satisfying a certain limit condition on f 2 , and tf pnqu´1 is meaning inversion modulo p.As an immediate application, we obtain results concerning the distribution of modular inverses inverses tf pnqu´1 pmod pq. The results apply, in particular, to Piatetski-Shapiro sequences tt c u with c P p1, 4 3 q. The proof is an adaptation of an argument used by Banks and the first named author in a series of papers from

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…we have T f pN, I q ą 0. Note that Corollary 1.3 is an analogue of [29,Theorem 5.1], where a result of this type is given for Beatty sequences.…”

Section: Corollary 13 Let F Be a Real-valued Twice-differentiable Fun...mentioning

confidence: 99%

Kloosterman sums with twice-differentiable functions

Shparlinski¹,

Technau²

2020

Funct. Approx. Comment. Math.

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“…In this regard it may also be worth pointing out that Theorem 1.1 yields good results for arithmetic functions like the divisor function, or Euler's totient, but may fail to provide useful information for certain other applications. The reader may imagine wanting to study S α (f, x), when f = χ is a Dirichlet character modulo q or something similar (see, for instance, [4,5,8], or [22] for a similar situation). Here the main focus usually lies in beating the trivial bound |S α (χ, x)| ≤ x.…”

Section: Introductionmentioning

confidence: 99%

Metric results on summatory arithmetic functions on Beatty sets

Technau¹,

Zafeiropoulos²

2019

Preprint

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Let f : N → C be an arithmetic function and consider the Beatty set B(α) = { nα : n ∈ N } associated to a real number α, where ξ denotes the integer part of a real number ξ. We show that the asymptotic formulaholds for almost all α > 1 with respect to the Lebesgue measure. This significantly improves an earlier result due to Abercrombie, Banks, and Shparlinski. The proof uses a recent Fourier-analytic result of Lewko and Radziwiłł based on the classical Carleson-Hunt inequality. Moreover, using a probabilistic argument, we establish the existence of functions f : N → { ±1 } for which the above error term is optimal up to logarithmic factors.

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Modular hyperbolas and Beatty sequences

Cited by 2 publications

References 0 publications

Kloosterman sums with twice-differentiable functions

Kloosterman sums with twice-differentiable functions

Metric results on summatory arithmetic functions on Beatty sets

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