Level of distribution of unbalanced convolutions

Abstract

We show that if an essentially arbitrary sequence supported on an interval containing x integers, is convolved with a tiny Siegel-Walfisz-type sequence supported on an interval containing exp((log x)^ε) integers then the resulting multiplicative convolution has (in a weak sense) level of distribution x^(1/2 + 1/66 − ε) as x goes to infinity. This dispersion estimate has a number of consequences for: the distribution of the kth divisor function to moduli x^(1/2 + 1/66 − ε) for any integer k ≥ 1, the distribution of products of exactly two primes in arithmetic progressions to large moduli, the distribution of sieve weights of level x^(1/2 + 1/66 − ε) to moduli as large as x^(1−ε) and for the Brun-Titchmarsh theorem for almost all moduli q of size x^(1−ε), lowering the long-standing constant 4 in that range. Our result improves and is inspired by earlier work of Green (and subsequent work of Granville-Shao) which is concerned with the distribution of 1-bounded multiplicative functions in arithmetic progressions to large moduli. As in these previous works the main technical ingredient are the recent estimates of Bettin-Chandee for trilinear forms in Kloosterman fractions and the estimates of Duke-Friedlander-Iwaniec for bilinear forms in Kloosterman fractions.