arXiv subjects

John T. Griesmer

Publications and source records attributed to John T. Griesmer.

At least 19 recordsLinked to original sources

arXiv2026

Bohr sets in sumsets III: expanding difference sets and almost Bohr sets

Let $G$ be a discrete abelian group. Følner showed that if $A \subseteq G$ has positive upper Banach density, then $A - A$ contains an almost Bohr set -- a set of the form $B \setminus E$ where $B$ is a Bohr set and $E$ has zero Banach density. We study the sets $S \subseteq G$ for which $A - A + S$ contains a Bohr set for every $A \subseteq G$ of positive upper Banach density. For $G = \mathbb{Z}$, we show that the sets $\{n^2: n \in \mathbb{N}\}$, $\{p - 1: p \text{ prime}\}$, and $\{ \lfloor n^c \rfloor: n \in \mathbb{N} \}$ with $c > 0$, have this property. Moreover, we prove that there are sets $A, B \subseteq \mathbb{Z}$ such that $A$ is dense in the Bohr topology of $\mathbb{Z}$, $d^*(B) > 0$, while $A + B$ is not piecewise Bohr, answering two questions of the second author in [31]. We also study those sets $S$ such that $A + S$ contains a Bohr set for every almost Bohr set $A$. As applications, we prove: (i) If $ϕ_1, ϕ_2: G \to G$ are (not necessarily commuting) homomorphisms with finite indices $[G: ϕ_i(G)]$, and $C \subseteq G$ is a central set, then $ϕ_1(C) - ϕ_1(C) + ϕ_2(C)$ contains a Bohr set. This answers one of our questions in [35] and generalizes results in [44, 48]; (ii) Every set of pointwise recurrence in $\mathbb{Z}$ is a set of nice recurrence and a van der Corput set, extending known properties of sets of pointwise recurrence studied in [26, 27, 40].

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arXiv2026

Kneser- and Jin-type inverse theorems in discrete abelian groups

We characterize the pairs of sets $A, B$ in an arbitrary (countable or uncountable) discrete abelian group $Γ$ satisfying $\tilde{m}(A+B)<\tilde{m}(A)+\tilde{m}(B)$, where $\tilde{m}$ is an arbitrary finitely additive translation-invariant probability measure on $Γ$, extending M.~Kneser's theorem on Haar measure in compact abelian groups. We then characterize, for an arbitrary Følner sequence or Følner net $\mathbf F=(F_{i})_{i\in I}$ on $Γ$, those $A$, $B$ satisfying $\underline{d}_{\mathbf F}(A+B)<\underline{d}_{\mathbf F}(A)+\underline{d}_{\mathbf F}(B)$, where $\underline{d}_{\mathbf F}(C):=\liminf_{i\in I} |C\cap F_{i}|/|F_{i}|$. This extends Kneser's theorem on lower asymptotic density in $\mathbb N$. We also generalize theorems of Prerna Bihani and Renling Jin by characterizing pairs $A$, $B$ satisfying $d^{*}(A+B)<d^{*}(A)+d^{*}(B)$, where $d^{*}$ is upper Banach density on $Γ$.

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arXiv2025

Discrete sumsets with one large summand

If $A$ and $B$ are subsets of an abelian group, their sumset is $A+B:=\{a+b:a\in A, b\in B\}$. We study sumsets in discrete abelian groups, where at least one summand has positive upper Banach density. Renling Jin proved that if $A$ and $B$ are sets of integers having positive upper Banach density, then $A+B$ is piecewise syndetic. Bergelson, Furstenberg, and Weiss improved the conclusion to "$A+B$ is piecewise Bohr." Beiglböck, Bergelson, and Fish showed this to be qualitatively optimal, in the sense that if $C\subseteq \mathbb Z$ is piecewise Bohr, then there are $A, B\subseteq \mathbb Z$ having positive upper Banach density such that $A+B\subseteq C$. We improve these results by establishing a strong correspondence between sumsets in discrete abelian groups, level sets of convolutions in compact abelian groups, and sumsets in compact abelian groups. Our proofs avoid measure preserving dynamics and nonstandard analysis, and our results apply to discrete abelian groups of any cardinality.

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arXiv2024

Separating measurable recurrence from strong recurrence via rigidity sequences

If $G$ is an abelian group, we say $S\subset G$ is a set of recurrence if for every probability measure preserving $G$-system $(X,μ,T)$ and every $D\subset X$ having $μ(D)>0$, there is a $g\in S$ such that $μ(D\cap T^{g}D)>0$. We say $S$ is a set of strong recurrence if for every set $D$ having $μ(D)>0$ there is a $c>0$ such that $μ(D\cap T^{g}D)>c$ for infinitely many $g\in S$. We call $S$ measure expanding if for all $g\in G$, the translate $S+g$ is a set of recurrence. A rigidity sequence for $(X,μ,T)$ is a sequence of elements $s_n\in G$ satisfying $\lim_{n\to\infty} μ(D\triangle T^{s_n}D)=0$ for all measurable $D\subset X$. For all but countably many countable abelian groups $G$, we prove that if $S$ is measure expanding, there is a sequence of elements $s_n\in S$ such that $\{s_n:n\in \mathbb N\}$ is also measure expanding and every translate of $(s_n)$ is a rigidity sequence for some free weak mixing measure preserving $G$-system. The special case where $S=G$ proves a conjecture of Ackelsberg. As a consequence, we prove that for every countably infinite abelian group $G$ and every measure expanding set $S\subset G$ there is a subset $S'\subset S$ such that $S'$ is measure expanding and no translate of $S'$ is a set of strong recurrence.

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arXiv2024

Separating topological recurrence from measurable recurrence: exposition and extension of Kriz's example

We prove that for every infinite set $E\subset \mathbb Z$, there is a set $S\subset E-E$ which is a set of topological recurrence and not a set of measurable recurrence. This extends a result of Igor Kriz, proving that there is a set of topological recurrence which is not a set of measurable recurrence. Our construction follows Kriz's closely, and this paper can be considered an exposition of the original argument.

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arXiv2024

Intersective sets for sparse sets of integers

For $E \subset \mathbb{N}$, a subset $R \subset \mathbb{N}$ is $E$-intersective if for every $A \subset E$ having positive upper relative density, we have $R \cap (A - A) \neq \varnothing$. On the other hand, $R$ is chromatically $E$-intersective if for every finite partition $E=\bigcup_{i=1}^k E_i$, there exists $i$ such that $R\cap (E_i-E_i)\neq\varnothing$. When $E=\mathbb{N}$, we recover the usual notions of intersectivity and chromatic intersectivity. In this article, we investigate to which extent known intersectivity results hold in the relative setting when $E = \mathbb{P}$, the set of primes, or other sparse subsets of $\mathbb{N}$. Among other things, we prove: -There exists an intersective set that is not $\mathbb{P}$-intersective. -However, every $\mathbb{P}$-intersective set is intersective. -There exists a chromatically $\mathbb{P}$-intersective set which is not intersective (and therefore not $\mathbb{P}$-intersective). -The set of shifted Chen primes $\mathbb{P}_{\mathrm{Chen}} + 1$ is $\mathbb{P}$-intersective (and therefore intersective).

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arXiv2023

Bohr sets in sumsets II: countable abelian groups

We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, letting $G$ be a countable discrete abelian group and $ϕ_1, ϕ_2, ϕ_3: G \to G$ be commuting endomorphisms whose images have finite indices, we show that (1) If $A \subset G$ has positive upper Banach density and $ϕ_1 + ϕ_2 + ϕ_3 = 0$, then $ϕ_1(A) + ϕ_2(A) + ϕ_3(A)$ contains a Bohr set. This generalizes a theorem of Bergelson and Ruzsa in $\mathbb{Z}$ and a recent result of the first author. (2) For any partition $G = \bigcup_{i=1}^r A_i$, there exists an $i \in \{1, \ldots, r\}$ such that $ϕ_1(A_i) + ϕ_2(A_i) - ϕ_2(A_i)$ contains a Bohr set. This generalizes a result of the second and third authors from $\mathbb{Z}$ to countable abelian groups. (3) If $B, C \subset G$ have positive upper Banach density and $G = \bigcup_{i=1}^r A_i$ is a partition, $B + C + A_i$ contains a Bohr set for some $i \in \{1, \ldots, r\}$. This is a strengthening of a theorem of Bergelson, Furstenberg, and Weiss. These results are quantitative in the sense that the radius and rank of the Bohr set obtained depends only on the indices $[G:ϕ_j(G)]$, the upper Banach density of $A$ (in (1)), or the number of sets in the given partition (in (2) and (3)).

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arXiv2023

A set of 2-recurrence whose perfect squares do not form a set of measurable recurrence

We say that $S\subset\mathbb Z$ is a set of $k$-recurrence if for every measure preserving transformation $T$ of a probability measure space $(X,μ)$ and every $A\subseteq X$ with $μ(A)>0$, there is an $n\in S$ such that $μ(A\cap T^{-n} A\cap T^{-2n}\cap \dots \cap T^{-kn}A)>0$. A set of $1$-recurrence is called a set of measurable recurrence. Answering a question of Frantzikinakis, Lesigne, and Wierdl, we construct a set of $2$-recurrence $S$ with the property that $\{n^2:n\in S\}$ is not a set of measurable recurrence.

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arXiv2022

Special cases and equivalent forms of Katznelson's problem on recurrence

We make three observations regarding a question popularized by Katznelson: is every subset of $\mathbb Z$ which is a set of Bohr recurrence is also a set of topological recurrence? (i) If $G$ is a countable abelian group and $E\subset G$ is an $I_0$ set, then every subset of $E-E$ which is a set of Bohr recurrence is also a set of topological recurrence. In particular every subset of $\{2^n-2^m : n,m\in \mathbb N\}$ which is a set of Bohr recurrence is a set of topological recurrence. (ii) Let $\mathbb Z^ω$ be the direct sum of countably many copies of $\mathbb Z$ with standard basis $E$. If every subset of $(E-E)-(E-E)$ which is a set of Bohr recurrence is also a set of topological recurrence, then every subset of every countable abelian group which is a set of Bohr recurrence is also a set of topological recurrence. (iii) Fix a prime $p$ and let $\mathbb F_p^ω$ be the direct sum of countably many copies of $\mathbb Z/p\mathbb Z$ with basis $(\mathbf e_i)_{i\in \mathbb N}$. If for every $p$-uniform hypergraph with vertex set $\mathbb N$ and edge set $\mathcal F$ having infinite chromatic number, the Cayley graph on $\mathbb F_p^ω$ determined by $\{\sum_{i\in F}\mathbf e_i:F\in \mathcal F\}$ has infinite chromatic number, then every subset of $\mathbb F_p^ω$ which is a set of Bohr recurrence is a set of topological recurrence.

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arXiv2021

Separating Bohr denseness from measurable recurrence

We prove that there is a set of integers $A$ having positive upper Banach density whose difference set $A-A:=\{a-b:a,b\in A\}$ does not contain a Bohr neighborhood of any integer, answering a question asked by Bergelson, Hegyvári, Ruzsa, and the author, in various combinations. In the language of dynamical systems, this result shows that there is a set of integers $S$ which is dense in the Bohr topology of $\mathbb Z$ and which is not a set of measurable recurrence. Our proof yields the following stronger result: if $S\subseteq \mathbb Z$ is dense in the Bohr topology of $\mathbb Z$, then there is a set $S'\subseteq S$ such that $S'$ is dense in the Bohr topology of $\mathbb Z$ and for all $m\in \mathbb Z,$ the set $(S'-m)\setminus \{0\}$ is not a set of measurable recurrence.

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arXiv2021

Bohr neighborhoods in generalized difference sets

If $A$ is a set of integers having positive upper Banach density and $r,s,t$ are nonzero integers whose sum is zero, a theorem of Bergelson and Ruzsa says that the set $rA+sA+tA:=\{ra_1+sa_2+ta_3:a_i\in A\}$ contains a Bohr neighborhood of zero. We prove the natural generalization of this result for subsets of countable abelian groups and more summands.

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arXiv2019

Semicontinuity of structure for small sumsets in compact abelian groups

We study pairs of subsets $A, B$ of a compact abelian group $G$ where the sumset $A+B:=\{a+b: a\in A, b\in B\}$ is small. Let $m$ and $m_{*}$ be Haar measure and inner Haar measure on $G$, respectively. Given $\varepsilon>0$, we classify all pairs $A,B$ of Haar measurable subsets of $G$ satisfying $m(A), m(B)>\varepsilon$ and $m_{*}(A+B)\leq m(A)+m(B)+δ$ where $δ=δ(\varepsilon)>0$ is small. We also study the case where the $δ$-popular sumset $A+_δB:=\{t\in G: m(A\cap (t-B))>δ\}$ is small. We prove that for all $\varepsilon>0$, there is a $δ>0$ such that if $A$ and $B$ are subsets of a compact abelian group $G$ having $m(A), m(B)>\varepsilon$ and $m(A+_δB)\leq m(A)+m(B)+δ$, then there are sets $S, T\subseteq G$ such that $m(A\triangle S)+m(B\triangle T)<\varepsilon$ and $m(S+T)\leq m(S)+m(T)$. Appealing to known results, the latter inequality yields strong structural information on $S$ and $T$, and therefore on $A$ and $B$.

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arXiv2018

Bohr sets in triple products of large sets in amenable groups

We answer a question of Hegyvári and Ruzsa concerning effective estimates of the Bohr-regularity of certain triple sums of sets with positive upper Banach densities in the integers. Our proof also works for any discrete amenable group, and it does not require all addends in the triple products we consider to have positive (left) upper Banach densities; one of the addends is allowed to only have positive upper asymptotic density with respect to a (possibly very sparse) ergodic sequence.

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arXiv2017

Bohr topology and difference sets for some abelian groups

For a fixed prime $p$, $\mathbb F_{p}$ denotes the field with $p$ elements, and $\mathbb F_{p}^ω$ denotes the countable direct sum $\bigoplus_{n=1}^{\infty} \mathbb F_{p}$. Viewing $\mathbb F_{p}^ω$ as a countable abelian group, we construct a set $A\subseteq \mathbb F_{p}^ω$ having positive upper Banach density while the difference set $A-A:=\{a-b:a,b\in A\}$ does not contain a Bohr neighborhood of any $c\in \mathbb F_{p}^ω$. For $p=2$ we obtain a stronger conclusion: $A-A$ does not contain a set of the form $g+(B-B)$, where $B$ is piecewise syndetic. This construction answers negatively a variant of the following question asked by several authors: if $A\subseteq \mathbb Z$ has positive upper Banach density, must $A-A$ contain a Bohr neighborhood of some $n\in \mathbb Z$? We also construct sets $S, A\subseteq \mathbb F_{p}^ω$ such that $S$ is dense in the Bohr topology of $\mathbb F_{p}^ω$, $A$ has positive upper Banach density, and $A+S$ is not piecewise Bohr. For $p=2$ we show that every translate of $S$ is a set of topological recurrence and $A+S$ is not piecewise syndetic. These constructions answer a variant of a question asked by the author.

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arXiv2017

Single recurrence in abelian groups

We collect problems on recurrence for measure preserving and topological actions of a countable abelian group, considering combinatorial versions of these problems as well. We solve one of these problems by constructing, in $G_{2}:=\bigoplus_{n=1}^{\infty} \mathbb Z/2\mathbb Z$, a set $S$ such that every translate of $S$ is a set of topological recurrence, while $S$ is not a set of measurable recurrence. This construction answers negatively a variant of the following question asked by several authors: if $A\subset \mathbb Z$ has positive upper Banach density, must $A-A$ contain a Bohr neighborhood of some $n\in \mathbb Z$? We also solve a variant of a problem posed by the author by constructing, for all $\varepsilon>0$, sets $S, A\subseteq G_{2}$ such that every translate of $S$ is a set of topological recurrence, $d^{*}(A)>1-\varepsilon$, and the sumset $S+A$ is not piecewise syndetic. Here $d^{*}$ denotes upper Banach density.

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arXiv2017

Bohr neighborhoods in three-fold difference sets

Answering a question of Hegyvári and Ruzsa , we show that if A is a set of integers having positive upper Banach density, then the set A+A-A:= {a+b-c: a, b, c are in A} contains Bohr neighborhoods of many elements of A, where the radius and dimension of the Bohr neighborhood depend only the upper Banach density of A.

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arXiv2016

Recurrence, rigidity, and popular differences

We construct a set $S$ such that every translate of $S$ is a set of recurrence and a set of rigidity for a weak mixing measure preserving system. This construction generalizes or strengthens results of Katznelson, Saeki, Forrest, and Fayad and Kanigowski. The construction provides a density analogue of Julia Wolf's results on popular differences in finite abelian groups.

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arXiv2013

An inverse theorem: when the measure of the sumset is the sum of the measures in a locally compact abelian group

We classify the pairs of subsets (A,B) of a locally compact abelian group satisfying m(A+B)=m(A)+m(B), where m is Haar measure. This generalizes a result of M. Kneser classifying such pairs under the additional assumption that G is compact and connected. Our proof combines Kneser's proof with arguments of D. Grynkiewicz, who classified the pairs of subsets (A,B) of abelian groups satisfying |A+B|=|A|+|B|, where |A| is the cardinality of A.

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