Algebra & Number Theory

Polynomial values modulo primes on average and sharpness of the larger sieve

Xuancheng Shao

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Abstract

This paper is motivated by the following question in sieve theory. Given a subset X [N] and α (0, 1 2). Suppose that |X(modp)| (α + o(1))p for every prime p. How large can X be? On the one hand, we have the bound |X|αNα from Gallagher’s larger sieve. On the other hand, we prove, assuming the truth of an inverse sieve conjecture, that the bound above can be improved (for example, to |X|αNO(α2014) for small α). The result follows from studying the average size of |X(modp)| as p varies, when X = f() [N] is the value set of a polynomial f(x) [x].

Article information

Source
Algebra Number Theory, Volume 9, Number 10 (2015), 2325-2346.

Dates
Received: 17 December 2014
Revised: 19 July 2015
Accepted: 17 August 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842448

Digital Object Identifier
doi:10.2140/ant.2015.9.2325

Mathematical Reviews number (MathSciNet)
MR3437764

Zentralblatt MATH identifier
1331.11083

Subjects
Primary: 11N35: Sieves
Secondary: 11R45: Density theorems 11R09: Polynomials (irreducibility, etc.)

Keywords
Gallagher's larger sieve inverse sieve conjecture value sets of polynomials over finite fields

Citation

Shao, Xuancheng. Polynomial values modulo primes on average and sharpness of the larger sieve. Algebra Number Theory 9 (2015), no. 10, 2325--2346. doi:10.2140/ant.2015.9.2325. https://projecteuclid.org/euclid.ant/1510842448


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