Andrew Granville's Home Page

2021 Publications

Large deviations of sums of random variables
(with Youness Lamzouri)
Lithunanian Mathematics Journal

In this paper, we investigate the large deviations of sums of weighted random variables that are approximately independent, generalizing and improving some of the results of Montgomery and Odlyzko. We are motivated by examples arising from number theory, including the sequences \( p^{it}, \chi(p), \chi_d(p), \lambda_f(p)\), and \( \text{Kl}_q(a-n, b)\); where \( p\) ranges over the primes, \( t\) varies in a large interval, \( \chi\) varies among all characters modulo \( q\), \( \chi_d\) varies over quadratic characters attached to fundamental discriminants \( |d|\leq x\), \( \lambda_f(n)\) are the Fourier coefficients of holomorphic cusp forms \( f\) of (a large) weight \( k\) for the full modular group, and \( \text{Kl}_q(a, b)\) are the normalized Kloosterman sums modulo a large prime \( q\), where \( a, b\) vary in \( (\mathbb{F}_q)^{\times}\).

Article and Paper

An alternative to Vaughan's identity
Rivista di Matematica della Università di Parma 12 (1) 119-124.

We exhibit an identity that plays the same role as Vaughan's identity but is arguably simpler.

Article and Journal

A tight structure theorem for sumsets
(with Aled Walker)
Proceedings of the American Mathematical Society 149 (10), 4073--4082.

Let \( A = \{0 = a_0 < a_1 < \cdots < a_{\ell + 1} = b\} \) be a finite set of non-negative integers. We prove that the sumset \( NA \) has a certain easily-described structure, provided that \( N \geqslant b-\ell \), as recently conjectured by Granville and Shakan. We also classify those sets \( A \) for which this bound cannot be improved.

Article and Reprint