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Random differences in Szemerédi’s theorem and related results. (English) Zbl 1414.11018
Summary: 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 \(\ell\) be a positive integer and \(\{{\mathbf u}_{1}\geq {\mathbf u}_{2}\geq \cdots\}\) be a decreasing sequence of probabilities satisfying \({\mathbf u}_{n} \cdot n^{1/(\ell+1)}\to \infty\). Let \(R = R^{\omega}\) 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 \(a, a+r, a+2r,\dotsc, a+\ell r\) of length \(\ell + 1\) with difference \(r\in R^{\omega}\). The best previous result (by M. Christ and us) was the condition \({\mathbf u}_{n}{\cdot}n^{2-\ell+1}\to \infty\) with a logarithmic rate. The new bound is better when \(\ell\geq 4\).
Our second improvement concerns almost everywhere convergence of double ergodic averages. We construct a (random) sequence \(\{r_{1} < r_{2} < \cdots \}\) of positive integers such that \(r_{n}/n^{2-\varepsilon}\to \infty\) for all \(\varepsilon > 0\) and, for any measure preserving transformation \(T\) of a probability space, the averages \[ \frac{1}{N}\sum_{n < N} {T^n}{F_1}(x) T^{r_n}F_2(x) \] converge for almost every \(x\). Our best previous result was the growth rate \(r_{n}/n^{(1+1/14)-\varepsilon}\to \infty\) of the sequence \(\{r_{n}\}\).

11B25 Arithmetic progressions
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
11B30 Arithmetic combinatorics; higher degree uniformity
05D40 Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.)
60F15 Strong limit theorems
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