## Algebra & Number Theory

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

Xuancheng Shao

#### 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 $X = f(ℤ) ∩ [N]$ is the value set of a polynomial $f(x) ∈ ℤ[x]$.

#### Article information

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

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

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.)

#### 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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