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

2014/09/25 by Xuancheng Shao, Shao, Xuancheng · 2 citations
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1409.7160

openalex publication_date 2014/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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\pmod p|≤ (α+o(1))p for every prime p. How large can X be? On the one hand, we have the bound |X|≪_αN^α 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|≪_αN^O(α²⁰¹⁴) for small α). The result follows from studying the average size of |X\pmod p| as p varies, when X=f(ℤ)∩ [N] is the value set of a polynomial f(x)∈ℤ[x].

Cited by

• Larger sieve with height function and uniform bounds for integral points on curves over number fields
• An inverse theorem on sets with rich additive structure modulo primes

Related

• On a variant of the large sieve
• On the exponential large sieve inequality for sparse sequences modulo primes
• Large sieve estimate for multivariate polynomial moduli and applications
• Inverse questions for the large sieve
• The large sieve for 2^[O(n^15/14+o(1))] modulo primes