arXiv subjects

Daniel Glasscock

Publications and source records attributed to Daniel Glasscock.

16 recordsLinked to original sources

arXiv2026

On sets of pointwise recurrence and dynamically thick sets

A set $A \subseteq \mathbb{N}$ is a set of pointwise recurrence if for all minimal dynamical systems $(X, T)$, all $x \in X$, and all open neighborhoods $U \subseteq X$ of $x$, there exists a time $n \in A$ such that $T^n x \in U$. The set $A$ is dynamically thick if the same holds for all non-empty, open sets $U \subseteq X$. Our main results give combinatorial characterizations of sets of pointwise recurrence and dynamically thick sets that allow us to answer questions of Host, Kra, Maass and Glasner, Tsankov, Weiss, and Zucker. We also introduce and study a local version of dynamical thickness called dynamical piecewise syndeticity. We show that dynamically piecewise syndetic sets are piecewise syndetic, generalizing results of Dong, Glasner, Huang, Shao, Weiss, and Ye. The proofs involve the algebra of families of large sets, dynamics on the space of ultrafilters, and our recent characterization of dynamically syndetic sets.

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arXiv2025

A structure theorem for polynomial return-time sets in minimal systems

We investigate the structure of return-time sets determined by orbits along polynomial tuples in minimal topological dynamical systems. Building on the topological characteristic factor theory of Glasner, Huang, Shao, Weiss, and Ye, we prove a structure theorem showing that, in a minimal system, return-time sets coincide -- up to a non-piecewise syndetic set -- with those in its maximal infinite-step pronilfactor. As applications, we establish three new multiple recurrence theorems concerning linear recurrence along dynamically defined syndetic sets and polynomial recurrence along arithmetic progressions in minimal and totally minimal systems. We also show how our main theorem can be used to prove that two previously separate conjectures -- one due to Glasner, Huang, Shao, Weiss, and Ye and the other due to Leibman -- are equivalent.

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arXiv2024

F\o{}lner, Banach, and translation density are equal and other new results about density in left amenable semigroups

In any semigroup $S$ satisfying the Strong Folner Condition, there are three natural notions of density for a subset $A$ of $S$: Folner density $d(A)$, Banach density $d^*(A)$, and translation density $d_t(A)$. If $S$ is commutative or left cancellative, it is known that these three notions coincide. We shall show that these notions coincide for every semigroup $S$ which satisfies the Strong Folner Condition. Using this fact, we solve a problem that has been open for decades, showing that the set of ultrafilters every member of which has positive Folner density is a two sided ideal of $\beta S$. We also show that, if $S$ is a left amenable semigroup, then the set of ultrafilters every member of which has positive Banach density is a two sided ideal of $\beta S$. We investigate the density properties of subsets of $S$ in the case in which the minimal left ideals of the Stone-\v{C}ech compactification $\beta S$ are singletons. This occurs in many familiar examples, including all semilattices and all semigroups which have a right zero. We show that this is equivalent to the statement that $S$ satisfies the Strong Folner Condition and that, for every subset $A$ of $S$, $d(A)\in \{0,1\}$. We also examine the relation between the density properties of two semigroups when one is a quotient of the other. The Folner density of a subset of $S$ is always determined by some Folner net in $S$. We show that an arbitrary Folner net in $S$ determines the density of all of the subsets of $S$. And we prove that, if $S$ and $T$ are left amenable semigroups, then $d^*(A\times B)=d^*(A)d^*(B)$ for every subset $A$ of $S$ and every subset $B$ of $T$.

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arXiv2024

Dynamically syndetic sets and the combinatorics of syndetic, idempotent filters

A subset of the positive integers is dynamically central syndetic if it contains the times that a point returns to a neighborhood of itself in a minimal topological dynamical system. These sets are part of the highly-influential link between dynamics and combinatorics forged by Furstenberg and Weiss in the 1970's. Our main result is a characterization of dynamically central syndetic sets as precisely those sets that belong to syndetic, idempotent filters. This gives a "global" analogue to the well-known "local" characterization of Furstenberg's central sets as members of minimal, idempotent ultrafilters. Applying the main result, we answer two open questions posed by Host, Kra, and Maass concerning sets of pointwise topological recurrence.

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arXiv2024

A combinatorial proof of a sumset conjecture of Furstenberg

We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if $\log r / \log s$ is irrational and $X$ and $Y$ are $\times r$- and $\times s$-invariant subsets of $[0,1]$, respectively, then $\dim_\text{H} (X+Y) = \min ( 1, \dim_\text{H} X + \dim_\text{H} Y)$. Our main result yields information on the size of the sumset $λX + ηY$ uniformly across a compact set of parameters at fixed scales. The proof is combinatorial and avoids the machinery of local entropy averages and CP-processes, relying instead on a quantitative, discrete Marstrand projection theorem and a subtree regularity theorem that may be of independent interest.

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arXiv2022

On Katznelson's Question for skew product systems

Katznelson's Question is a long-standing open question concerning recurrence in topological dynamics with strong historical and mathematical ties to open problems in combinatorics and harmonic analysis. In this article, we give a positive answer to Katznelson's Question for certain towers of skew product extensions of equicontinuous systems, including systems of the form $(x,t) \mapsto (x + α, t + h(x))$. We describe which frequencies must be controlled for in order to ensure recurrence in such systems, and we derive combinatorial corollaries concerning the difference sets of syndetic subsets of the natural numbers.

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arXiv2022

Simultaneous approximation in nilsystems and the multiplicative thickness of return-time sets

In the topological dynamical system $(X,T)$, a point $x$ simultaneously approximates a point $y$ if there exists a sequence $n_1$, $n_2$, ... of natural numbers for which $T^{n_i} x$, $T^{2n_i}x$, ..., $T^{k n_i} x$ all tend to $y$. In 1978, Furstenberg and Weiss showed that every system possesses a point which simultaneously approximates itself (a multiply recurrent point) and deduced refinements of van der Waerden's theorem on arithmetic progressions. In this paper, we study the denseness of the set of points that are simultaneously approximated by a given point. We show that in a minimal nilsystem, all points simultaneously approximate a $\delta$-dense set of points under a necessarily restricted set of powers of $T$. We tie this theorem to the multiplicative combinatorial properties of return-time sets, showing that all nil-Bohr sets and typical return-time sets in a minimal system are multiplicatively thick in a coset of a multiplicative subsemigroup of the natural numbers. This yields an inhomogeneous multiple recurrence result that generalizes Furstenberg and Weiss' theorem and leads to new enhancements of van der Waerden's theorem. This work relies crucially on continuity in the prolongation relation (the closure of the orbit-closure relation) developed by Auslander, Akin, and Glasner; the theory of rational points and polynomials on nilmanifolds developed by Leibman, Green, and Tao; and the machinery of topological characteristic factors developed recently by Glasner, Huang, Shao, Weiss, and Ye.

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arXiv2020

Additive and geometric transversality of fractal sets in the integers

By juxtaposing ideas from fractal geometry and dynamical systems, Furstenberg proposed a series of conjectures in the late 1960's that explore the relationship between digit expansions with respect to multiplicatively independent bases. In this work, we introduce and study - in the discrete context of the integers - analogues of some of the notions and results surrounding Furstenberg's work. In particular, we define a new class of fractal sets of integers that parallels the notion of $\times r$-invariant sets on the 1-torus and investigate the additive and geometric independence between two such fractal sets when they are structured with respect to multiplicatively independent bases. Our main results in this direction parallel the works of Furstenberg, Hochman-Shmerkin, Shmerkin, Wu, and Lindenstrauss-Meiri-Peres and include: -a classification of all subsets of the positive integers that are simultaneously $\times r$- and $\times s$-invariant; -integer analogues of two of Furstenberg's transversality conjectures pertaining to the dimensions of the intersection $A\cap B$ and the sumset $A+B$ of $\times r$- and $\times s$-invariant sets $A$ and $B$ when $r$ and $s$ are multiplicatively independent; and -a description of the dimension of iterated sumsets $A+A+\cdots+A$ for any $\times r$-invariant set $A$. We achieve these results by combining ideas from fractal geometry and ergodic theory to build a bridge between the continuous and discrete regimes. For the transversality results, we rely heavily on quantitative bounds on the $L^q$-dimensions of projections of restricted digit Cantor measures obtained recently by Shmerkin. We end by outlining a number of open questions and directions regarding fractal subsets of the integers.

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arXiv2019

Norm forms represent few integers but relatively many primes

Norm forms, examples of which include $x^2 + y^2$, $x^2 + x y - 57 y^2$, and $x^3 + 2 y^3 + 4 z^3 - 6 x y z$, are integral forms arising from norms on number fields. We prove that the natural density of the set of integers represented by a norm form is zero, while the relative natural density of the set of prime numbers represented by a norm form exists and is positive. These results require tools from class field theory, including the Artin-Chebotarev density theorem and the Hilbert class field. We introduce these tools as we need them in the course of the main arguments. This article is expository in nature and assumes only a first course in algebraic number theory.

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arXiv2019

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.

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arXiv2018

A Khintchine-type theorem and solutions to linear equations in Piatetski-Shapiro sequences

Our main result concerns a perturbation of a classic theorem of Khintchine in Diophantine approximation. We give sufficient conditions on a sequence of positive real numbers $(ψ_n)_{n \in \mathbb{N}}$ and differentiable functions $(φ_n: J \to \mathbb{R})_{n \in \mathbb{N}}$ so that for Lebesgue-a.e. $θ\in J$, the inequality $\| nθ+ φ_n(θ) \| \leq ψ_n$ has infinitely many solutions. The main novelty is that the magnitude of the perturbation $|φ_n(θ)|$ is allowed to exceed $ψ_n$, changing the usual "shrinking targets" problem into a "shifting targets" problem. As an application of the main result, we prove that if the linear equation $y=ax+b$, $a, b \in \mathbb{R}$, has infinitely many solutions in $\mathbb{N}$, then for Lebesgue-a.e. $α> 1$, it has infinitely many or finitely many solutions of the form $\lfloor n^α\rfloor$ according as $α< 2$ or $α> 2$.

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arXiv2016

Shift equivalence in the generalized factor order

We provide a geometric condition that guarantees strong Wilf equivalence in the generalized factor order. This provides a powerful tool for proving specific and general Wilf equivalence results, and several such examples are given.

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arXiv2016

What is a graphon?

Graphons, short for graph functions, are limiting objects for sequences of large, finite graphs with respect to the so-called cut metric. In this expository piece, we define graphons, motivate them, and discuss how they complete the space of finite graphs. We conclude by stating three theorems that connect the finite world of graphs with the continuous world of graphons.

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arXiv2016

Multiplicative richness of additively large sets in $\mathbb{Z}^d$

In their proof of the IP Szemerédi theorem, a far reaching extension of the classic theorem of Szemerédi on arithmetic progressions, Furstenberg and Katznelson introduced an important class of additively large sets called $\text{IP}_{\text{r}}^*$ sets which underlies recurrence aspects in dynamics and is instrumental to enhanced formulations of combinatorial results. The authors recently showed that additive $\text{IP}_{\text{r}}^*$ subsets of $\mathbb{Z}^d$ are multiplicatively rich with respect to every multiplication on $\mathbb{Z}^d$ without zero divisors (e.g. multiplications induced by degree $d$ number fields). In this paper, we explain the relationships between classes of multiplicative largeness with respect to different multiplications on $\mathbb{Z}^d$. We show, for example, that in contrast to the case for $\mathbb{Z}$, there are infinitely many different notions of multiplicative piecewise syndeticity for subsets of $\mathbb{Z}^d$ when $d \geq 2$. This is accomplished by using the associated algebra representations to prove the existence of sets which are large with respect to some multiplications while small with respect to others. In the process, we give necessary and sufficient conditions for a linear transformation to preserve a class of multiplicatively large sets. One consequence of our results is that additive $\text{IP}_{\text{r}}^*$ sets are multiplicatively rich in infinitely many genuinely different ways. We conclude by cataloging a number of sources of additive $\text{IP}_{\text{r}}^*$ sets from combinatorics and dynamics.

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arXiv2016

On the interplay between notions of additive and multiplicative largeness and its combinatorial applications

Many natural notions of additive and multiplicative largeness arise from results in Ramsey theory. In this paper, we explain the relationships between these notions for subsets of $\mathbb{N}$ and in more general ring-theoretic structures. We show that multiplicative largeness begets additive largeness in three ways and give a collection of examples demonstrating the optimality of these results. We also give a variety of applications arising from the connection between additive and multiplicative largeness. For example, we show that given any $n, k \in \mathbb{N}$, any finite set with fewer than $n$ elements in a sufficiently large finite field can be translated so that each of its elements becomes a non-zero $k^{\text{th}}$ power. We also prove a theorem concerning Diophantine approximation along multiplicatively syndetic subsets of $\mathbb{N}$ and a theorem showing that subsets of positive upper Banach density in certain multiplicative sub-semigroups of $\mathbb{N}$ of zero density contain arbitrarily long arithmetic progressions. Along the way, we develop a new characterization of upper Banach density in a wide class of amenable semigroups and make explicit the uniformity in recurrence theorems from measure theoretic and topological dynamics. This in turn leads to strengthened forms of classical theorems of Szemer\'edi and van der Waerden on arithmetic progressions.

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arXiv2016

Solutions to certain linear equations in Piatetski-Shapiro sequences

Denote by $\text{PS}(α)$ the image of the Piatetski-Shapiro sequence $n \mapsto \lfloor n^α \rfloor$ where $α> 1$ is non-integral and $\lfloor x \rfloor$ is the integer part of $x \in \mathbb{R}$. We partially answer the question of which bivariate linear equations have infinitely many solutions in $\text{PS}(α)$: if $a, b \in \mathbb{R}$ are such that the equation $y=ax+b$ has infinitely many solutions in the positive integers, then for Lebesgue-a.e. $α> 1$, it has infinitely many or at most finitely many solutions in $\text{PS}(α)$ according as $α< 2$ (and $0 \leq b < a$) or $α> 2$ (and $(a,b) \neq (1,0)$). We collect a number of interesting open questions related to further results along these lines.

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