Gaussian Width Bounds with Applications to Arithmetic Progressions in Random Settings

Abstract

Motivated by a problem on random differences in Szemerédi's theorem and another problem on large deviations for arithmetic progressions in random sets, we prove upper bounds on the Gaussian width of special point sets in R k . The point sets are formed by the image of the n-dimensional Boolean hypercube under a mapping ψ : R n → R k , where each coordinate is a constant-degree multilinear polynomial with 0-1 coefficients. We show the following applications of our bounds. Let [Z/N Z] p be the random subset of Z/N Z containing each element independently with probability p.This gives a polynomial improvement for all ℓ ≥ 3 of a previous bound due to Frantzikinakis, Lesigne and Wierdl, and reproves more directly the same improvement shown recently by the authors and Dvir (here we avoid the theories of locally decodable codes and quantum information).• Let X k be the number of k-term arithmetic progressions in [Z/N Z] p and consider the large deviation rate ρ k (δ) = log Pr[X k ≥ (1 + δ)EX k ]. We give quadratic improvements of the best-known range of p for which a highly precise estimate of ρ k (δ) due to Bhattacharya, Ganguly, Shao and Zhao is valid for all odd k ≥ 5. In particular, the estimate holds if p ≥ ω(N −c k log N ) for c k = (6k⌈(k − 1)/2⌉) −1 .We also discuss connections with locally decodable codes and the Banach-space notion of type for injective tensor products of ℓ p -spaces.

Upper Tail Large Deviations for Arithmetic Progressions in a Random Set

“…For fixed δ > 0, the theorem requires p to decay slower than N − 1 6k(k−1) . Very recently, Briët and Gopi [5] improved the exponent from 1 6k(k−1) to 1 6k (k−1)/2 , which improves our result for all k ≥ 5. It remains an open problem to extend the range of validity of p. In comparison, Warnke's asymptotics (1.1) on the order of log-probability holds for all p ≥ (log N/N ) 1/(k−1) .…”

Bivariate fluctuations for the number of arithmetic progressions in random sets

“…Complementing these results, Bhattacharya, Ganguly, Shao, and Zhao [2] pinned down the precise large deviation rate function for "sufficiently large" p. By contrast to the approach in [27], the proof in [2] builds on the non-linear large deviation principle by Chatterjee and Dembo [8] and its refinement due to Eldan [11] in terms of the concept of Gaussian width, a particular notion of complexity. Recently, Briët and Gopi [6] derived an upper bound on the Gaussian width leading to an improvement of the lower bound on p given in [2]. The special case ℓ = 3 was already included in [8].…”

Untitled

“…) is a ±1-valued random variable with expected value α. 8 Finally, we use the smoothness property to transform the f i into decoders with the desired properties. This is done in Section 3.…”

“…However, standard arguments show that the two statements are equivalent 6. In[8], the first and third authors derive Corollary 1.7 in addition to another result in probabilistic combinatorics from first principles (which is to say, not through LDCs).…”