arXiv subjects

Sigrid Grepstad

Publications and source records attributed to Sigrid Grepstad.

16 recordsLinked to original sources

arXiv2025

Bounded remainder sets, bounded distance equivalent cut-and-project sets, and equidecomposability

We use the measurable Hall's theorem due to Cie\'sla and Sabok to prove that (i) if two measurable sets $A,B \subset \mathbb{R}^d$ of the same measure are bounded remainder sets with respect to a totally irrational $d$-dimensional vector $\alpha$, then $A, B$ are equidecomposable with measurable pieces using translations from $\mathbb{Z} \alpha + \mathbb{Z}^d$; and (ii) given a lattice $\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$ with projections $p_1$ and $p_2$ onto $\mathbb{R}^m$ and $\mathbb{R}^n$ respectively, if two cut-and-project sets in $\mathbb{R}^m$ obtained from Riemann measurable windows $W, W' \subset \mathbb{R}^n$ are bounded distance equivalent, then $W, W'$ are equidecomposable with measurable pieces using translations from $p_2(\Gamma)$. We also prove by a different method that for one-dimensional cut-and-project sets, if the windows $W, W' \subset \mathbb{R}^n$ are polytopes then the pieces can also be chosen to be polytopes; however this result fails in dimensions two and higher.

math.MG↗

arXiv2025

Bounded lattice tiles that pack with another lattice

Suppose L and M are full-rank lattices in Euclidean space, such that vol(L) < vol(M). Answering a question of Han and Wang from 2001, we show how to construct a bounded measurable set F (we can even take F to be a finite union of polytopes) such that F+L is a tiling and F+M is a packing. If we do not require measurability of F it is often possible that a set F can be found tiling with both L and M even when L and M have different volumes, for instance if their intersection is trivial. We also show here that such a set can never be bounded if L and M have different volumes.

math.CA↗

arXiv2025

Bounded common fundamental domains for two lattices

We prove that for any two lattices $L, M \subseteq \mathbb{R}^d$ of the same volume there exists a measurable, bounded, common fundamental domain of them. In other words, there exists a bounded measurable set $E \subseteq \mathbb{R}^d$ such that $E$ tiles $\mathbb{R}^d$ when translated by $L$ or by $M$. In fact, the set $E$ can be taken to be a finite union of polytopes. A consequence of this is that the indicator function of $E$ forms a Weyl--Heisenberg (Gabor) orthogonal basis of $L^2(\mathbb{R}^d)$ when translated by $L$ and modulated by $M^*$, the dual lattice of $M$.

math.CA↗

arXiv2024

Bounded distance equivalence of cut-and-project sets and equidecomposability

We show that given a lattice $\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$, and projections $p_1$ and $p_2$ onto $\mathbb{R}^m$ and $\mathbb{R}^n$ respectively, cut-and-project sets obtained using Jordan measurable windows $W$ and $W'$ in $\mathbb{R}^n$ of equal measure are bounded distance equivalent only if $W$ and $W'$ are equidecomposable by translations in $p_2(\Gamma)$. As a consequence, we obtain an explicit description of the bounded distance equivalence classes in the hulls of simple quasicrystals. A corrigendum is appended at the end of the paper.

math.DS↗

arXiv2022

Hilbert points in Hilbert space-valued $L^p$ spaces

Let $H$ be a Hilbert space and $(\Omega,\mathcal{F},\mu)$ a probability space. A Hilbert point in $L^p(\Omega; H)$ is a nontrivial function $\varphi$ such that $\|\varphi\|_p \leq \|\varphi+f\|_p$ whenever $\langle f, \varphi \rangle = 0$. We demonstrate that $\varphi$ is a Hilbert point in $L^p(\Omega; H)$ for some $p\neq2$ if and only if $\|\varphi(\omega)\|_H$ assumes only the two values $0$ and $C>0$. We also obtain a geometric description of when a sum of independent Rademacher variables is a Hilbert point.

math.FA↗

arXiv2021

On the order of magnitude of Sudler products II

We study the asymptotic behavior of Sudler products $P_N(\alpha)= \prod_{r=1}^{N}2|\sin \pi r\alpha|$ for quadratic irrationals $\alpha \in \mathbb{R}$. In particular, we verify the convergence of certain perturbed Sudler products along subsequences, and show that $\liminf_N P_N(\alpha) = 0$ and $\limsup_N P_N(\alpha)/N = \infty$ whenever the maximal digit in the continued fraction expansion of $\alpha$ exceeds $23$. This generalizes results obtained for the period one case $\alpha=[0; \overline{a}]$.

math.NT↗

arXiv2020

F. Wiener's trick and an extremal problem for $H^p$

For $0<p \leq \infty$, let $H^p$ denote the classical Hardy space of the unit disc. We consider the extremal problem of maximizing the modulus of the $k$th Taylor coefficient of a function $f \in H^p$ which satisfies $\|f\|_{H^p}\leq1$ and $f(0)=t$ for some $0 \leq t \leq 1$. In particular, we provide a complete solution to this problem for $k=1$ and $0<p<1$. We also study F. Wiener's trick, which plays a crucial role in various coefficient-related extremal problems for Hardy spaces.

math.CV↗

arXiv2019

On the asymptotic behaviour of the sine product $\prod_{r=1}^n|2\sin(πr α)|$

In this paper we review recently established results on the asymptotic behaviour of the trigonometric product $P_n(α) = \prod_{r=1}^n |2\sin πr α|$ as $n\to \infty$. We focus on irrationals $α$ whose continued fraction coefficients are bounded. Our main goal is to illustrate that when discussing the regularity of $P_n(α)$, not only the boundedness of the coefficients plays a role; also their size, as well as the structure of the continued fraction expansion of $α$, is important.

math.NT↗

arXiv2018

A positive lower bound for $\liminf_{N\to\infty} \prod_{r=1}^N \left| 2\sin \pi r \varphi \right|$

Nearly 60 years ago, Erd\H{o}s and Szekeres raised the question of whether $$\liminf_{N\to \infty} \prod_{r=1}^N \left| 2\sin \pi r \alpha \right| =0$$ for all irrationals $\alpha$. Despite its simple formulation, the question has remained unanswered. It was shown by Lubinsky in 1999 that the answer is yes if $\alpha$ has unbounded continued fraction coefficients, and it was suggested that the answer is yes in general. However, we show in this paper that for the golden ratio $\varphi=(\sqrt{5}-1)/2$, $$\liminf_{N\to \infty} \prod_{r=1}^N \left| 2\sin \pi r \varphi \right| >0 ,$$ providing a negative answer to this long-standing open problem.

math.NT↗

arXiv2018

Asymptotic behaviour of the Sudler product of sines for quadratic irrationals

We study the asymptotic behaviour of the sequence of sine products $P_n(α) = \prod_{r=1}^n |2\sin πr α|$ for real quadratic irrationals $α$. In particular, we study the subsequence $Q_n(α)=\prod_{r=1}^{q_n} |2\sin πr α|$, where $q_n$ is the $n$th best approximation denominator of $α$, and show that this subsequence converges to a periodic sequence whose period equals that of the continued fraction expansion of $α$. This verifies a conjecture recently posed by Mestel and Verschueren.

math.NT↗

arXiv2017

On pair correlation and discrepancy

We say that a sequence $\{x_n\}_{n \geq 1}$ in $[0,1)$ has Poissonian pair correlations if \begin{equation*} \lim_{N \rightarrow \infty} \frac{1}{N} \# \left\{ 1 \leq l \neq m \leq N \, : \, \left\lVert x_l-x_m \right\rVert < \frac{s}{N} \right\} = 2s \end{equation*} for all $s>0$. In this note we show that if the convergence in the above expression is - in a certain sense - fast, then this implies a small discrepancy for the sequence $\{x_n\}_{n \geq 1}$. As an easy consequence it follows that every sequence with Poissonian pair correlations is uniformly distributed in $[0,1)$.

math.NT↗

arXiv2016

Riesz bases, Meyer's quasicrystals, and bounded remainder sets

We consider systems of exponentials with frequencies belonging to simple quasicrystals in $\mathbb{R}^d$. We ask if there exist domains $S$ in $\mathbb{R}^d$ which admit such a system as a Riesz basis for the space $L^2(S)$. We prove that the answer depends on an arithmetical condition on the quasicrystal. The proof is based on the connection of the problem to the discrepancy of multi-dimensional irrational rotations, and specifically, to the theory of bounded remainder sets. In particular it is shown that any bounded remainder set admits a Riesz basis of exponentials. This extends to several dimensions (and to the non-periodic setting) the results obtained earlier in dimension one.

math.CA↗

arXiv2016

Sets of bounded remainder for the continuous irrational rotation on $[0,1)^2$

We study sets of bounded remainder for the two-dimensional continuous irrational rotation $(\{x_1+t\}, \{x_2+tα\})_{t \geq 0}$ in the unit square. In particular, we show that for almost all $α$ and every starting point $(x_1, x_2)$, every polygon $S$ with no edge of slope $α$ is a set of bounded remainder. Moreover, every convex set $S$ whose boundary is twice continuously differentiable with positive curvature at every point is a bounded remainder set for almost all $α$ and every starting point $(x_1, x_2)$. Finally we show that these assertions are, in some sense, best possible.

math.NT↗

arXiv2016

Spectra for cubes in products of finite cyclic groups

We consider "cubes" in products of finite cyclic groups and we study their tiling and spectral properties. (A set in a finite group is called a tile if some of its translates form a partition of the group and is called spectral if it admits an orhogonal basis of characters for the functions supported on the set.) We show an analog of a theorem due to Iosevich and Pedersen, Lagarias, Reeds and Wang, and the third author of this paper, which identified the tiling complements of the unit cube in Euclidean space with the spectra of the same cube.

math.CA↗

arXiv2014

Sets of bounded discrepancy for multi-dimensional irrational rotation

We study bounded remainder sets with respect to an irrational rotation of the $d$-dimensional torus. The subject goes back to Hecke, Ostrowski and Kesten who characterized the intervals with bounded remainder in dimension one. First we extend to several dimensions the Hecke-Ostrowski result by constructing a class of $d$-dimensional parallelepipeds of bounded remainder. Then we characterize the Riemann measurable bounded remainder sets in terms of "equidecomposability" to such a parallelepiped. By constructing invariants with respect to this equidecomposition, we derive explicit conditions for a polytope to be a bounded remainder set. In particular this yields a characterization of the convex bounded remainder polygons in two dimensions. The approach is used to obtain several other results as well.

math.DS↗

arXiv2013

Multi-tiling and Riesz bases

Let S be a bounded, Riemann measurable set in R^d, and L be a lattice. By a theorem of Fuglede, if S tiles R^d with translation set L, then S has an orthogonal basis of exponentials. We show that, under the more general condition that S multi-tiles R^d with translation set L, S has a Riesz basis of exponentials. The proof is based on Meyer's quasicrystals.

math.CA↗