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**1 - 4**of**4**### 3 MULTIPLE RECURRENCE FOR NON-COMMUTING TRANSFORMATIONS ALONG RATIONALLY INDEPENDENT POLYNOMIALS

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### RANDOM DIFFERENCES IN SZEMERÉDI’S THEOREM AND RELATED RESULTS

"... ABSTRACT. We introduce a new, elementary method for studying random differences in arithmetic progres-sions and convergence phenomena along random sequences of integers. We apply our method to obtain sig-nificant improvements on two results. The first improvement is the following: Let ` be a positiv ..."

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ABSTRACT. We introduce a new, elementary method for studying random differences in arithmetic progres-sions and convergence phenomena along random sequences of integers. We apply our method to obtain sig-nificant 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) →∞. Let R = Rω be the ran-dom sequence we get by selecting the natural number n with probability un. If A is a set of natural numbers with positive upper 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