Equidistribution from the Chinese Remainder Theorem

Kowalski, Emmanuel; Soundararajan, K.

doi:10.1016/j.aim.2021.107776

Cited by 7 publications

(6 citation statements)

References 11 publications

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“…A key feature of the exponential sums considered above is their twisted multiplicativity. In this section we formulate, following Hooley [18], Fouvry and Michel [12], and our own recent paper [27], a general result on bounding averages of twisted multiplicative functions.…”

Section: Sums Of Twisted-multiplicative Functionsmentioning

confidence: 99%

Exponential Sums, Twisted Multiplicativity, and Moments

Kowalski¹,

Soundararajan²

2022

Analysis at Large

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Section: Sums Of Twisted-multiplicative Functionsmentioning

confidence: 99%

Exponential Sums, Twisted Multiplicativity, and Moments

Kowalski¹,

Soundararajan²

2022

Analysis at Large

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“…This is a significant difference from the one-dimensional situation, where it is conjectured that for any irreducible polynomial of degree greater than one, that if we restrict to prime moduli, then the roots modulo p are still uniformly distributed, as has been proven for the quadratic case by Duke, Friedlander and Iwaniec [3], see also [9]. In a forthcoming paper [6], Kowalski and Soundararajan prove a very general result that is closely related to ours when restricted to squarefree moduli.…”

Section: Introductionmentioning

confidence: 98%

On the Joint Distribution of the Roots of Pairs of Polynomial Congruences

Zehavi¹

2020

Preprint

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Let f (x) ∈ Z[x] be a primitive irreducible polynomial of degree greater than one. In [5], Hooley showed that the sequence µ n , where f (µ) = 0(n), ordered in the obvious way, is uniformly distributed modulo one. It is the goal of this paper to show that if f (x), g(x) ∈ Z[x] are a pair of primitive irreducible polynomials of degree greater than one, not necessarily distinct, then the sequence ( µ n , ν n ), with f (µ) = 0(n) and g(ν) = 0(n), ordered in the obvious way, is uniformly distributed modulo one in the unit torus.

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“…Note that an integer-valued polynomial is not necessarily a pseudo-polynomial as X(X+ 1)/2 shows, see below. Pseudo-polynomials have long been studied for themselves, but they have also found recent applications in analytic number theory [9,10].…”

Section: Introductionmentioning

confidence: 99%

“…Note that if P (X) ∈ C[X] is such that the sequence (P (n)) n≥0 is a pseudo-polynomial, then P (X) ∈ Q[X], but P (X) does not necessarily belong to Z[X] as P (X) = 1 2 X(X + 1) shows. Pseudo-polynomials have long been studied for themselves, but they have also found recent applications in analytic number theory [8,9].…”

Section: Introductionmentioning

confidence: 99%

On primary pseudo-polynomials (around Ruzsa’s conjecture)

Delaygue

Rivoal

2022

Int. J. Number Theory

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Every polynomial [Formula: see text] satisfies the congruences [Formula: see text] for all integers [Formula: see text]. An integer valued sequence [Formula: see text] is called a pseudo-polynomial when it satisfies these congruences. Hall characterized pseudo-polynomials and proved that they are not necessarily polynomials. A long-standing conjecture of Ruzsa says that a pseudo-polynomial [Formula: see text] is a polynomial as soon as [Formula: see text]. Under this growth assumption, Perelli and Zannier proved that the generating series [Formula: see text] is a [Formula: see text]-function. A primary pseudo-polynomial is an integer valued sequence [Formula: see text] such that [Formula: see text] for all integers [Formula: see text] and all prime numbers [Formula: see text]. The same conjecture has been formulated for them, which implies Ruzsa’s, and this paper revolves around this conjecture. We obtain a Hall type characterization of primary pseudo-polynomials. We give a new proof and generalize a result due to Zannier that any primary pseudo-polynomial with an algebraic generating series is a polynomial. We make the Perelli–Zannier Theorem effective and we prove a Pólya type result.

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Equidistribution from the Chinese Remainder Theorem

Cited by 7 publications

References 11 publications

Exponential Sums, Twisted Multiplicativity, and Moments

Exponential Sums, Twisted Multiplicativity, and Moments

On the Joint Distribution of the Roots of Pairs of Polynomial Congruences

On primary pseudo-polynomials (around Ruzsa’s conjecture)

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