Motivated by two problems on arithmetic progressions (APs)—concerning large
deviations for AP counts in random sets and random differences in Szemer´edi’s theorem—
we prove upper bounds on the Gaussian width of the image of the n-dimensional Boolean
hypercube under a mapping ψ : Rn → Rk, where each coordinate is a constant-degree
multilinear polynomial with 0/1 coefficients. We show the following applications of our
bounds. Let [Z/NZ]p be the random subset of Z/NZ containing each element independently
with probability p.
• Let Xk be the number of k-term APs in [Z/NZ]p. We show that a precise estimate
on the large deviation rate log Pr[Xk ≥ (1 + δ)EXk] due to Bhattacharya, Ganguly,
Shao and Zhao is valid if