A Hardy field extension of Szemerédi's theorem (Q2389236)

In 1975 \textit{E. Szemerédi} [Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)] proved that a set of integers of positive upper density contains arbitrary long arithmetic progression which was previously conjectured by Erdös and Turán. After Szemerédi's proof of this theorem, there have been a lot of investigation on possible forms of the common difference \(d\) of the arithmetic progressions in Szemerédi's theorem (see the reference in the paper). Most notably is a result of \textit{V. Bergelson} and \textit{A. Leibman} [J. Am. Math. Soc. 9, No.~3, 725--753 (1996; Zbl 0870.11015)], who showed that \(d\) can be taken to be of the form \(p(n)\) where \(p\) is any non-constant integer polynomial with \(p(0) = 0\). In this paper under review, the authors extend Bergelson and Leibman's result to a larger class of functions that belong to some Hardy field with some growth conditions. To state the results, let \(B\) be the set of equivalence classes of real valued function \(a(x)\) defined on some open half line \((u, \infty),\) where two functions are in the same equivalence class if they agree for all large \(x.\) A Hardy field is a subfield of the ring \((B, +, \cdot)\) that is closed under differentiation. Let \({\mathcal H}\) denote the union of all Hardy fields. The first main result (Theorem A) of the paper is the following. Let \(a \in {\mathcal H}\) satisfy \(x^k \prec a(x) \prec x^{k+1}\) for some non-negative integer \(k.\) Let \(\ell \in {\mathbb N}.\) Then for every \(\Lambda \subset {\mathbb Z}\) with positive upper density contains arithmetic progression of the form \(\{m, m + [a(n)], m + 2 [a(n)], \ldots, m + \ell [a(n)]\}\) for some \(m\in {\mathbb Z}\) and \(n \in {\mathbb N}\) with \(a(n) \neq 0\) where \([x]\) denotes the integer part of \(x.\) Suppose that for every \(m \in {\mathbb N}\) the set \(S \subset {\mathbb Z}\) contains arithmetic progressions of the form \(\{c_m + m n : 1 \leq n \leq N_m\}\) where \(c_m, N_m \) are integers with \(N_m \to \infty.\) Let \(\ell \in {\mathbb N}.\) Then the second main result (Theorem~B) of the paper asserts that every subset of integers \(\Lambda\) with positive upper density contains arithmetic progressions of the form \(\{r, r+s, r + 2 s, \ldots, r + \ell s\}\) for some \(r \in {\mathbb Z}\) and non-zero \(s \in S.\) The main idea of the proof is first to use the correspondence principle of \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] to reformulate the above mentioned results (Theorem A and B) as corresponding statements (Theorem A\('\) and B\('\)) about multiple recurrences in ergodic theory so that techniques from ergodic theory can be applied. There are three major parts in the proof: (i) The authors show that the range of \([a(n)]\) contains some suitably chosen polynomial patterns of fixed degree for \(a \in {\mathcal H}\) satisfying the required growth conditions and then work with this collection of polynomial patterns. (ii) For the polynomial patterns found in (i), there's a naturally associated multiple ergodic averages. The authors reduce the problem to establishing a certain multiple recurrence property for nilsystems. (iii) The multiple recurrence property for nilsystems can be verified by comparing the multiple ergodic average along the polynomial patterns found in (i) with some easier to handle averages that can be estimated using Furstenberg's classical multiple recurrence result. A quantitative equidistribution result which was recently obtained by Green and Tao is used to carry out the comparison step.