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 \( (\mathbb{F}_q)^{\times}\).

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

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.