Random differences in Szemerédi’s theorem and related results

Abstract

We introduce a new, elementary method for studying random differences in arithmetic progressions and convergence phenomena along random sequences of integers. We apply our method to obtain significant improvements on two results. The first improvement is the following. Let l be a positive integer and {u ₁ ≥ u ₂ ≥ · · · } be a decreasing sequence of probabilities satisfying u _n · n ^1/(l+1)→∞. Let R = R ^ω be the random sequence obtained by selecting the natural number n with probability u _n . Then every set A of natural numbers with positive upper density contains an arithmetic progression a, a+r, a+2r,..., a+lr of length l + 1 with difference r ∈ R ^ω. The best previous result (by M. Christ and us) was the condition u _n ·n ^2−l+1 → ∞ with a logarithmic rate. The new bound is better when l ≥ 4. Our second improvement concerns almost everywhere convergence of double ergodic averages. We construct a (random) sequence {r ₁ < r ₂ < · · · } of positive integers such that r _n /n ^2−ϵ → ∞ for all ϵ > 0 and, for any measure preserving transformation T of a probability space, the averages

$$\frac{1}{N}\sum\limits_{n < N} {{T^n}{F_1}\left( x \right)} {T^{{r_n}}}{F_2}\left( x \right)$$

converge for almost every x. Our best previous result was the growth rate r _n /n ^{(1+1/14)−ϵ} → ∞ of the sequence {r _n }.