arXiv subjects

Luke Hetzel

Publications and source records attributed to Luke Hetzel.

3 recordsLinked to original sources

arXiv2026

Periodic structure and Schrodinger operators for codings of circle rotations

We consider, for any irrational $\alpha$ and interval $I \subset \mathbb{T}$, the 2-interval coding subshift $X^{(I, \alpha)}$ induced by coding orbits under repeated rotation by $\alpha$ via membership in $I$ or $I^c$. Each sequence $c \in X^{(I, \alpha)}$ has an associated Schr\"{o}dinger operator $H_c$, and in \cite{kaminaga} it was proved that if the continued fraction of $\alpha$ has digits with limsup at least $4$, then almost every $c \in X^{(I, \alpha)}$ has so-called $3$-block Gordon structure, which implies that the operator $H_c$ has no eigenvalues. We significantly improve this result by completely characterizing almost-sure $3$-block Gordon structure, proving that in fact it holds for all $(\alpha,I)$ except for a countable set of pairs $(\alpha, |I|)$ where $\alpha$ is M\"{o}bius equivalent to the silver mean and $|I| \in \mathbb{Z}\alpha + f(\alpha)$ where $f(\alpha)$ is a specific infinite series taking value either $\frac{1}{2}, \frac{\alpha}{2}$, or $\frac{\alpha+1}{2}$. We also show that for a set of $\alpha$ of full measure, and for every $I$, the set of points whose orbit codings do not have $3$-block Gordon structure has Hausdorff dimension bounded away from $1$.

math.DS↗

arXiv2026

On growth rates of infinite and finite sumsets

We study growth rates of infinite and finite sumset patterns in sets of positive density. In the infinite setting, we show that no such rate exists, answering a question of Kra, Moreira, Ritcher, and Robertson. Namely, for any proposed growth rate $\mathcal{H}: \mathbb{N} \to \mathbb{N}$ tending to infinity, we construct a set $A$ of lower density $1$ such that whenever $B,C \subseteq \mathbb{N}$ are infinite and $B+C \subseteq A$ we have the minimum of $|B\cap [N]|$ and $|C \cap [N]|$ is less than $\mathcal{H}(N)$ for infinitely many $N$. In the finitary setting, we prove that for all $\delta \in (0,1)$, for all sufficiently large $N$, for all subsets $A$ of $\{1,\dots,N\}$ of proportion $\delta$, one can always find sumset patterns $B+C\subseteq A$ with $|B|$ and $|C|$ of order $\log N$, partially resolving a conjecture of Kra, Moreira, Richter, and Robertson. Moreover, we generalize our second result to the case of the $k$-fold sum $B_1 + B_2 + \ldots + B_k \subseteq A$.

math.CO↗

arXiv2025

On infinite sumsets and sets of multiple recurrence

We answer two questions of Kra, Moreira, Richter and Robertson regarding the existence of infinite sumsets of the form $B + C$ in dense and sparse sets of integers and the relation of sumsets to sets of recurrence. We then further generalize these results, yielding new characterizations of sets of multiple measurable and topological recurrence.

math.DS↗