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 ` be a positive integer and let u1 ≥u2 ≥... be a decreasing sequence of probabilities satisfying un ·n1/(`+1) →∞. Select the natural number n into a random sequence R = Rω of integers with density un. Let A be a set of natural numbers with positive density. Then A contains an arithmetic progression a, a+r, a+2r,..., a+`r of length `+1 with difference r ∈Rω. The best earlier result (by M. Christ and us) was the condition un ·n2−`+1 → ∞ with a logarithmic speed. The new bound is better when ` ≥ 4. The other improvement concerns almost everywhere convergence of double ergodic averages: we (ran-domly) construct a sequence r1 < r2 <... of positive integers so that for any > 0 we have rn /n2−→ ∞ and for any measure preserving transformation T of a probability space the averages (0.1