Inverse questions for the large sieve

Let $A\subset [1,N]$ be a set of positive integers with $|A|\gg \sqrt N$. We show that if avoids about $p/2$ residue classes modulo $p$ for each prime $p$, the $A$ must correlate additively with the squares $S=\{n^2:1\leq n\leq \sqrt N\}$, in the sense that we have the additive energy estimate $E(A,S)\gg N\log N$. This is, in a sense, optimal.

We introduce a variant of the large sieve and give an example of its use in a sieving problem. Take the interval [N] = {1,...,N} and, for each odd prime p <= N^{1/2}, remove or ``sieve out'' by all n whose reduction mod p lies in some interval I_p of Z/pZ of length (p-1)/2. Let A be the set that remains: then |A| << N^{1/3 + o(1)}, a bound which improves slightly on the bound of |A| << N^{1/2} which results from applying the large sieve in its usual form. This is a very, very...

This paper is motivated by the following question in sieve theory. Given a subset $X\subset [N]$ and $\alpha\in (0,1/2)$. Suppose that $|X\pmod p|\leq (\alpha+o(1))p$ for every prime $p$. How large can $X$ be? On the one hand, we have the bound $|X|\ll_{\alpha}N^{\alpha}$ 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 $|X|\ll_{\alpha}N^{O(\alpha^{2014})}$ for ...

Extending a method of D. Wolke, we establish a general result on the large sieve with sparse sets S of moduli which are in a sense well-distributed in arithmetic progressions. We then use this result together with Fourier techniques to obtain large sieve bounds for the case when S consists of squares. These bounds improve a recent result by L. Zhao.

We prove a lower bound for the large sieve with square moduli.

In this paper we study the distribution of squares modulo a square-free number $q$. We also look at inverse questions for the large sieve in the distribution aspect and we make improvements on existing results on the distribution of $s$-tuples of reduced residues.

In this paper, we develop a large sieve type inequality with characters to square moduli. One expects that the result should be weaker than the classical inequality, but, conjecturally at least, not by much. The method is generalizable to higher power moduli.

Extending a method of D. Wolke, we establish a general result on the large sieve with sparse sets S of moduli which are in a sense well-distributed in arithmetic progressions. We then apply our result to the case when S consists of sqares. In this case we obtain an estimate which improves a recent result by L. Zhao.

We prove an estimate for the large sieve with square moduli which improves a recent result of L. Zhao. Our method uses an idea of D. Wolke and some results from Fourier analysis.

For a set of primes $\mathcal{P}$, let $\Psi(x, \mathcal{P})$ be the number of positive integers $n \leq x$ all of whose prime factors lie in $\mathcal{P}$. In this paper we classify the sets of primes $\mathcal{P}$ such that $\Psi(x, \mathcal{P})$ is within a constant factor of its expected value. This task was recently initiated by Granville, Koukoulopoulos and Matom\"aki and their main conjecture is proved in this paper. In particular our main theorem implies that, if not ...