Large subsets of discrete hypersurfaces in Zd\mathbb{Z}^d contain arbitrarily many collinear points

In 1977 L.T. Ramsey showed that any sequence in $\mathbb{Z}^2$ with bounded gaps contains arbitrarily many collinear points. Thereafter, in 1980, C. Pomerance provided a density version of this result, relaxing the condition on the sequence from having bounded gaps to having gaps bounded on average. We give a higher dimensional generalization of these results. Our main theorem is the following. Theorem: Let $d\in\mathbb{N}$, let $f:\mathbb{Z}^d\to\mathbb{Z}^{d+1}$ be a Lipschitz map and let $A\subset\mathbb{Z}^d$ have positive upper Banach density. Then $f(A)$ contains arbitrarily many collinear points. Note that Pomerance's theorem corresponds to the special case $d=1$. In our proof, we transfer the problem from a discrete to a continuous setting, allowing us to take advantage of analytic and measure theoretic tools such as Rademacher's theorem.Comment: 16 pages, small part of the argument clarified in light of suggestions from the refere

Multiplicative combinatorial properties of return time sets in minimal dynamical systems

We investigate the relationship between the dynamical properties of minimal topological dynamical systems and the multiplicative combinatorial properties of return time sets arising from those systems. In particular, we prove that for a residual sets of points in any minimal system, the set of return times to any non-empty, open set contains arbitrarily long geometric progressions. Under the separate assumptions of total minimality and distality, we prove that return time sets have positive multiplicative upper Banach density along $\mathbb{N}$ and along multiplicative subsemigroups of $\mathbb{N}$, respectively. The primary motivation for this work is the long-standing open question of whether or not syndetic subsets of the positive integers contain arbitrarily long geometric progressions; our main result is some evidence for an affirmative answer to this question.Comment: 32 page

Single and multiple recurrence along non-polynomial sequences

We establish new recurrence and multiple recurrence results for a rather large family $\mathcal{F}$ of non-polynomial functions which includes tempered functions defined in [11], as well as functions from a Hardy field with the property that for some $\ell\in \mathbb{N}\cup\{0\}$, $\lim_{x\to\infty }f^{(\ell)}(x)=\pm\infty$ and $\lim_{x\to\infty }f^{(\ell+1)}(x)=0$. Among other things, we show that for any $f\in\mathcal{F}$, any invertible probability measure preserving system $(X,\mathcal{B},\mu,T)$, any $A\in\mathcal{B}$ with $\mu(A)>0$, and any $\epsilon>0$, the sets of returns $$R_{\epsilon, A}= \big\{n\in\mathbb{N}:\mu(A\cap T^{-\lfloor f(n)\rfloor}A)>\mu^2(A)-\epsilon\big\}$$ and $$R^{(k)}_{A}= \big\{ n\in\mathbb{N}: \mu\big(A\cap T^{\lfloor f(n)\rfloor}A\cap T^{\lfloor f(n+1)\rfloor}A\cap\cdots\cap T^{\lfloor f(n+k)\rfloor}A\big)>0\big\}$$ possess somewhat unexpected properties of largeness; in particular, they are thick, i.e., contain arbitrarily long intervals.Comment: 51 page

Disjointness for measurably distal group actions and applications

We generalize Berg's notion of quasi-disjointness to actions of countable groups and prove that every measurably distal system is quasi-disjoint from every measure preserving system. As a corollary we obtain easy to check necessary and sufficient conditions for two systems to be disjoint, provided one of them is measurably distal. We also obtain a Wiener--Wintner type theorem for countable amenable groups with distal weights and applications to weighted multiple ergodic averages and multiple recurrence.Comment: 28 page