Multiple recurrence and large intersections for abelian group actions, Discrete Analysis 2021:18, 91 pp. In 1975, Szemerédi proved his famous theorem that asserts that for every positive integer $k$ and every $\delta>0$ there exists $n$ such that every subset of $\{1,2,\dots,n\}$ of size at least $\delta n$ contains an arithmetic progression of length $k$. A couple of years later, Furstenberg found a second proof of the theorem using ergodic theory. An advantage of Furstenberg's approach was that it was more amenable to generalization, a result of which was that it led to a whole new field of combinatorial ergodic theory and the discovery of important results such as the density Hales-Jewett theorem, due to Furstenberg and Katznelson, and the polynomial Szemerédi theorem of Bergelson and Leibman. The starting point for the ergodic approach to these problems is a general principle known as the Furstenberg correspondence, which makes it possible to translate many combinatorial problems into equivalent statements about topological or measure-preserving dynamical systems. For example, Szemerédi's theorem itself is equivalent to the following statement. Let $X$ be a probability space and $T:X\to X$ a measure-preserving map (meaning that $|T^{-1}(A)|=|A|$ for every measurable subset $A$ of $X$). Then for every $k$ and every $A\subset X$ of positive measure there exists $n$ such that $$|A\cap T^{-n}(A)\cap T^{-2n}(A)\dots\cap T^{-(k-1)n}(A)|>0.$$ It is natural to ask whether something more precise can be said. In particular, can we find some explicit constant $c>0$, depending on $k$ and $|A|$, and show that the lim sup of the measure on the left-hand side is at least $c$? And if so, what $c$ can we hope to obtain? A random set of density $\delta$ (or rather, a dynamical system built out of a sequence of such sets via the Furstenberg correspondence) shows easily that one cannot hope for better than $|A|^k$. And the following simple argument shows that one can attain a bound of $|A|^k$ when $k=2$. Let $A$ be a set of measure $\delta$ and let $\epsilon>0$. Let $f$ be the sum of the characteristic functions of $A, T^{-1}A, \dots, T^{-(r-1)}A$, and let us write $A_i$ for $T^{-i}A$. Then $\|f\|_1=r\delta$, which implies by Cauchy-Schwarz that $\|f\|_2\geq r\delta$. But $\|f\|_2^2=\sum_{i,j=0}^r|A_i\cap A_j|$, which implies that $\sum_{i\ne j}|A_i\cap A_j|\geq r^2\delta^2-r\delta$. By averaging, for large $r$ this gives us $i\ne j$ such that $|A_i\cap A_j|\geq\delta^2-\epsilon$, and then by the measure-preserving property we obtain $n$ such that $|A\cap T^{-n}A|\geq\delta^2-\epsilon$. This argument can also be used to show that the set $R$ of $n$ such that $|A\cap T^{-n}A|\geq\delta^2-\epsilon$ is _syndetic_, which means that there is some $N$ such that $R$ intersects every interval of length $N$ (or in other words it has _bounded gaps_). Indeed, if that is not the case, then for any $r$ we can inductively build a sequence $n_1,n_2,\dots,n_r$ such that none of the sums $n_{i_1}+\dots+n_{i_t}$ belong to $R$. But by the above argument there must exist $i0$ and $A\subset X$ is a measurable set, then the set $\{g\in G:|A\cap T_{\phi(g)}^{-1}A\cap T_{\psi(g)}^{-1}A|\geq|A|^2-\epsilon\}$ is syndetic. They also obtain some related combinatorial results. For instance, the construction of Ruzsa mentioned above is of a subset of $\{1,2,\dots,n\}$ of density $\exp(-c\sqrt{\log n})$ that contains no non-degenerate subset of the form $\{q(0),q(1),q(2),q(3),q(4)\}$ where $q$ is a polynomial of degree at most 2. (Note that if we replace 4 by 3 and "degree at most 2" by "degree 1" then we obtain a definition of an arithmetic progression of length 4.) Given a quintuple $r=(r_0,\dots,r_4)$, the authors define a _quadratic $r$-configuration_ of length 5 to be a set of the form $\{q(r_0),\dots,q(r_4)\}$. They then show that for every $r$ there is a constant $c(r)$ such that for each $n$ there is a subset of $\{1,2,\dots,n\}$ of density at least $\exp(-c(r)\sqrt{\log n})$ that contains no such configuration. This turns out to imply the failure of the large intersection property in the following situation: we have a dynamical system indexed by $\mathbb Z$ and distinct positive integers $r_1,\dots,r_4$, where we regard $r_i$ as the homomorphism that multiplies $\mathbb Z$ by $r_i$. As well as these and other results, the paper contains some interesting open problems. For example, they conjecture that, more generally, five genuinely distinct homomorphisms cannot have the large-intersections property. More precisely, if $\phi_1,\dots,\phi_4$ are homomorphisms from $G$ to $G$ such that any two of them are equal only on a subgroup of infinite index (and the same is true of any one of them and the identity), then the Khintchine recurrence theorem fails, meaning that we can find a measure preserving system indexed by $G$ and a set $A$ of positive measure such that the set $\{g\in G:|A\cap T_{\phi_1(g)}^{-1}A\cap \dots\cap T_{\phi_4(g)}^{-1}A|\geq|A|^5-\epsilon\}$ is not syndetic. The result previously mentioned proves this when $G=\mathbb Z$.