Gaussian width bounds with applications to arithmetic progressions in random settings
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 p ≥ ω(N−ck logN) for ck = (6k⌈(k−1)/2⌉)−1, which slightly improves their bound of ck = (6k(k − 1))−1 for k ≥ 5 (and matching their c3 and c4). • A set D ⊆ Z/NZ is ℓ-intersective if every dense subset of Z/NZ contains a non-trivial (ℓ+1)-term AP with common difference in D. We show that [Z/NZ]p is ℓ-intersective with probability 1 − oN(1) provided p ≥ ω(N−βℓ logN) for βℓ = (⌈(ℓ + 1)/2⌉)−1, improving the bound βℓ = (ℓ+1)−1 due to Frantzikinakis, Lesigne andWierdl for ℓ ≥ 2 and reproving more directly the same result shown recently by the authors and Dvir. In addition, we discuss some intriguing connections with special kinds of error correcting codes (locally decodable codes) and the Banach-space notion of type for injective tensor products of ℓp-spaces.