arXiv subjects

Sinai Robins

Publications and source records attributed to Sinai Robins.

At least 19 recordsLinked to original sources

arXiv2026

A note on a sparse sampling conjecture

A conjecture made by the second author was that two convex, centrally symmetric bodies of positive measure which are not multi-tilers must agree up to a rigid motion whenever the Fourier transforms of their indicator functions agree on $\Z^d$. Counterexamples were constructed by the first author in dimensions $d\geq4$, showing that the lattice $\Z^d$ can be too sparse. Here we show that the initial conjecture is also false in the remaining dimensions $2$ and $3$. However, we prove a related positive result, with a stronger conclusion and without either central symmetry or the non-multitiling assumption. Let $\mathcal L\subset\R^d$ be a full-rank lattice, and let $\mathcal P,Q\subset\R^d$ be connected finite unions of convex bodies such that no two distinct points of either set are congruent modulo $\mathcal L$. If the Fourier transforms of their indicator functions agree on the dual lattice $\mathcal L^*$, then $Q=\mathcal P+\ell$ for some $\ell\in \mathcal L$. In particular, if both sets are symmetric about the origin, then $\mathcal P=Q$.

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arXiv2026

The integer point enumerator of one irrational translate of P is a complete invariant

For a full-dimensional rational polytope $P\subset\mathbb{R}^d$ and a real dilation parameter $t>0$, the integer point enumerator is defined by $L_{P}(t):= |tP\cap\mathbb{Z}^d|$. We determine exactly which translation vectors $\mathbf y=(y_1,\ldots,y_d)\in\mathbb{R}^d$ have the property that the single translated counting function $t\longmapsto L_{P+\mathbf y}(t)$, with $t\in\mathbb{Q}_{>0}$, uniquely determines $P$ among all full-dimensional rational polytopes in $\mathbb{R}^d$. The necessary and sufficient condition is that $1,y_1,\ldots,y_d$ be linearly independent over $\mathbb{Q}$. In particular, we may use the explicit algebraic vector $\mathbf y^* := (2^{1/(d+1)},2^{2/(d+1)},\ldots,2^{d/(d+1)})$ in every dimension $d$. The sufficiency proof recovers the primitive facet inequalities from isolated discontinuities of the counting function, while necessity follows from an affine-unimodular obstruction.

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arXiv2026

Half-open integer parallelepipeds and polytope Dedekind sums

We study the Ehrhart theory of half-open $d$-dimensional integer parallelepipeds $\Pi$. Although the lattice-point count $t\Pi\cap \Z^d$ is known to be simply $\vol \Pi t^d$ for positive integer $t$, the corresponding counting function for arbitrary real dilations $t$ has subtle, nontrivial periodic structure. We give explicit formulas for this real Ehrhart quasi-polynomial, and more generally for all the discrete moments of the real dilates of $\Pi$: $\sum_{p\in t\Pi\cap\mathbb Z^d}\langle p,z\rangle^m$. The formulas are expressed in terms of Barnes polynomials and polytope Dedekind sums, which encode the periodic lattice flow of translated integer lattices on the flat torus determined by $\Pi$. Our approach develops further the study of polytope Dedekind sums, introduced recently in \cite{Robins2026}. In particular, we obtain novel identities for polytope Dedekind sums by using iterated discrete derivatives. Moreover, we show that the Ehrhart quasi-coefficients of $L_\Pi(t)$ are precisely alternating sums of polytope Dedekind sums. Finally, we give an Ehrhart-type reciprocity law relating $L_{\Pi}(t)$ at negative arguments to the lattice-point count of the `opposite' half-open parallelepiped.

math.CO↗

arXiv2026

Ehrhart quasi-polynomials via Barnes polynomials and discrete moments of parallelepipeds

We give novel explicit formulas for the Ehrhart quasi-polynomials of any rational polytope $P$, in terms of Barnes polynomials and discrete moments of half-open parallelepipeds. These formulas hold for all positive dilations of $P$ and involve an auxiliary complex $z$-parameter that yields compact formulations. We give analogous formulas for discrete moments of $P$ and its positive dilates, closely related to the literature on sums of polynomials over a polytope. Barnes polynomials and Barnes numbers permit explicit computations, showing that the main complexity in computing Ehrhart quasi-polynomials lies in the discrete moments of half-open integer parallelepipeds that we study here. These moments are generally summed over a particular lattice flow on a compact torus, introduced below. Developing these moments further, we interpret them as geometric higher-dimensional Dedekind sums, which we call polytope Dedekind sums, and establish reciprocity laws extending the classical Dedekind sum reciprocity law. As an application, we obtain a novel vertex formula for the general codimension-$1$ quasi-coefficient $c_{d-1}(t)$, valid for all $t>0$. Our main results also yield novel canonical vanishing identities for a rational polytope $P$, extending the well-known identities of Brion--Vergne. As a further consequence, we obtain a differential equation for discrete moments of $P$, extending work of Eva Linke. For smooth polytopes, we obtain novel and substantially simpler formulas for Ehrhart polynomials, discrete moments, and vanishing identities. From the perspective of Barnes polynomials and Barnes numbers, these identities may be of independent interest. They also demonstrate the utility of Barnes polynomials in geometric combinatorics, through their rich structure that extends the $1$-dimensional Bernoulli polynomials.

math.CO↗

arXiv2024

Sharp inequalities for discrete and continuous multi-tiling, using the Bombieri-Siegel approach

Given a finite subset $F$ of integer points in $\mathbb Z^d$, it is of interest to seek conditions on $F$ that allow it to multi-tile $\mathbb Z^d$ by translations. To this end, we give a discretized version of the Bombieri-Siegel formula, which represents a finite sum of discrete covariograms in terms of Fourier transforms. As a consequence, we arrive at a new equivalent condition for multi-tiling $\mathbb Z^d$ by translating $F$ with a fixed integer sublattice. In the continuous case, we study lattice sums of the cross covariogram for any two bounded sets $A, B\subset \mathbb R^d$, and we prove a refined continuous version of the classical Bombieri-Siegel formula from the geometry of numbers. To achieve this goal, we use a variant of the Poisson Summation formula, adapted for continuous functions of compact support. As an application of this refined Bombieri-Siegel formula, a new characterization of multi-tilings of Euclidean space by translations of a compact set by using a lattice is given. One consequence is a novel spectral formula for the volume of any bounded measurable set. Another consequence is a novel spectral formula for the product of the volumes of any two bounded measurable sets.

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arXiv2024

Rational eigenfunctions of the Hecke operators

We study the action of the Hecke operators $U_n$ on the space $\mathcal R$ of rational functions in one variable, over $\mathbb C$. The main goal is to give a complete classification of the eigenfunctions of $U_n$. We accomplish this by introducing certain number-theoretic directed graphs, called Zolotarev Graphs, which extend the well-known permutations due to Zolotarev. We develop the theory of these Zolotarev graphs, using them to decompose the eigenfunctions of $U_n$ into certain natural finite-dimensional vector spaces of rational functions, which we call the eigenspaces. In this context, we prove that the dimension of each eigenspace is equal to the number of nodes of a cycle that belongs to its corresponding Zolotarev graph. We prove that the number of leaves of this Zolotarev graph equals the dimension of the kernel of $U_n$. We then give a novel number-theoretic formula for the number of cycles of fixed length, in each Zolotarev graph. We also study the simultaneous eigenfunctions for all of the $U_n$, and give explicit bases for all of them. In the process, we answer many questions that were set out in the work of Gil and Robins (2005). We also discover certain strong relations between these graphs and the kernel of $U_n$ acting on a subspace of $\mathcal R$; in particular, we give several equivalent conditions for the diagonalizibility of $U_n$. Finally, we prove that the classical Artin Conjecture on primitive roots is equivalent to a new conjecture here, that infinitely many of these eigenspaces have dimension $1$.

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arXiv2023

The integer point transform as a complete invariant

The integer point transform $σ_{\mathcal P}$ is an important invariant of a rational polytope $\mathcal P$, and here we show that it is a complete invariant. We prove that it is only necessary to evaluate $σ_{\mathcal P}$ at one algebraic point in order to uniquely determine $\mathcal P$, by employing the Lindemann-Weierstrass theorem. Similarly, we prove that it is only necessary to evaluate the Fourier transform of a rational polytope $\mathcal P$ at a single algebraic point, in order to uniquely determine $\mathcal P$. We prove that identical uniqueness results also hold for integer cones. In addition, by relating the integer point transform to finite Fourier transforms, we show that a finite number of \emph{integer point evaluations} of $σ_{\mathcal P}$ suffice in order to uniquely determine $\mathcal P$. We also give an equivalent condition for central symmetry of a finite point set, in terms of the integer point transform, and prove some facts about its local maxima. Most of the results are proven for arbitrary finite sets of integer points in $\mathbb R^d$.

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arXiv2023

A friendly introduction to Fourier analysis on polytopes

This book is an introduction to the nascent field of Fourier analysis on polytopes, and cones. There is a rapidly growing number of applications of these methods, so it is appropriate to invite students, as well as professionals, to the field. Of the many applications of these techniques, we have chosen to focus on the following topics: (a) Formulations for the Fourier transform of a polytope (b) Minkowski and Siegel's theorems in the geometry of numbers (c) Tilings and multi-tilings of Euclidean space by translations of a polytope (d) Computing discrete volumes of polytopes, which are combinatorial approximations to the continuous volume (e) Sphere packings, and their packing density (f) Iterating the divergence theorem to give new formulations for the Fourier transform of a polytope, with applications (g) Shannon sampling, in several variables (h) More topics in the classical geometry of numbers We assume familiarity with Linear Algebra, with some Calculus and infinite series. Throughout, we introduce the topics gently, by giving many examples and exercises, so that this book is ideally suited for a course, or for self-study.

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arXiv2022

Coefficients of the solid angle and Ehrhart quasi-polynomials

Macdonald studied a discrete volume measure for a rational polytope $P$, called solid angle sum, that gives a natural discrete volume for $P$. We give a local formula for the codimension two quasi-coefficient of the solid angle sum of $P$. We also show how to recover the classical Ehrhart quasi-polynomial from the solid angle sum and in particular we find a similar local formula for the codimension one and codimension two quasi-coefficients. These local formulas are naturally valid for all positive real dilates of $P$. An interesting open question is to determine necessary and sufficient conditions on a polytope $P$ for which the discrete volume of $P$ given by the solid angle sum equals its continuous volume: $A_P(t) = \mathrm{vol}(P) t^d$. We prove that a sufficient condition is that $P$ tiles $\mathbb R^d$ by translations, together with the Hyperoctahedral group.

math.CO↗

arXiv2021

On the period collapse of a family of Ehrhart quasi-polynomials

A graph whose nodes have degree 1 or 3 is called a $\{1,3\}$-graph. Liu and Osserman associated a polytope to each $\{1,3\}$-graph and studied the Ehrhart quasi-polynomials of these polytopes. They showed that the vertices of these polytopes have coordinates in the set $\{0,\frac14,\frac12,1\}$, which implies that the period of their Ehrhart quasi-polynomials is either 1, 2, or 4. We show that the period of the Ehrhart quasi-polynomial of these polytopes is at most 2 if the graph is a tree or a cubic graph, and it is equal to 4 otherwise. In the process of proving this theorem, several interesting combinatorial and geometric properties of these polytopes were uncovered, arising from the structure of their associated graphs. The tools developed here may find other applications in the study of Ehrhart quasi-polynomials and enumeration problems for other polytopes that arise from graphs. Additionally, we have identified some interesting connections with triangulations of 3-manifolds.

math.CO↗

arXiv2021

The null set of a polytope, and the Pompeiu property for polytopes

We study the null set $N(\mathcal{P})$ of the Fourier-Laplace transform of a polytope $\mathcal{P} \subset \mathbb{R}^d$, and we find that $N(\mathcal{P})$ does not contain (almost all) circles in $\mathbb{R}^d$. As a consequence, the null set does not contain the algebraic varieties $\{z \in \mathbb{C}^d \mid z_1^2 + \dots + z_d^2 = α^2\}$ for each fixed $α\in \mathbb{C}$, and hence we get an explicit proof that the Pompeiu property is true for all polytopes. Our proof uses the Brion-Barvinok theorem, which gives a concrete formulation for the Fourier-Laplace transform of a polytope, and it also uses properties of Bessel functions. The original proof that polytopes (as well as other bodies) possess the Pompeiu property was given by Brown, Schreiber, and Taylor (1973) for dimension 2. Williams (1976) later observed that the same proof also works for $d>2$ and, using eigenvalues of the Laplacian, gave another proof valid for $d \geq 2$ that polytopes have the Pompeiu property.

math.MG↗

arXiv2020

An Euler-Maclaurin formula for polygonal sums

We prove an Euler-Maclaurin formula for double polygonal sums and, as a corollary, we obtain approximate quadrature formulas for integrals of smooth functions over polygons with integer vertices. Our Euler-Maclaurin formula is in the spirit of Pick's theorem on the number of integer points in an integer polygon and involves weighted Riemann sums, using tools from Harmonic analysis. Finally, we also exhibit a classical trick, dating back to Huygens and Newton, to accelerate convergence of these Riemann sums.

math.CA↗

arXiv2019

Spherical tetrahedra with rational volume, and spherical Pythagorean triples

We study spherical tetrahedra with rational dihedral angles and rational volumes. Such tetrahedra occur in the Rational Simplex Conjecture by Cheeger and Simons, and we supply vast families, discovered by computational efforts, of positive examples that confirm this conjecture. As a by-product, we also obtain a classification of all spherical Pythagorean triples, previously found by Smith.

math.MG↗

arXiv2019

Dragging the roots of a polynomial to the unit circle

Several conditions are known for a self-inversive polynomial that ascertain the location of its roots, and we present a framework for comparison of those conditions. We associate a parametric family of polynomials $p_α$ to each such polynomial $p$, and define $\mathscr{cn}(p)$, $\mathscr{il}(p)$ to be the sharp threshold values of $α$ that guarantee that, for all larger values of the parameter, $p_α$ has, respectively, all roots in the unit circle and all roots interlacing the roots of unity of the same degree. Interlacing implies circle rootedness, hence $\mathscr{il}(p)\geq\mathscr{cn}(p)$, and this inequality is often used for showing circle rootedness. Both $\mathscr{cn}(p)$ and $\mathscr{il}(p)$ turn out to be semi-algebraic functions of the coefficients of $p$, and some useful bounds are also presented, entailing several known results about roots in the circle. The study of $\mathscr{il}(p)$ leads to a rich classification of real self-inversive polynomials of each degree, organizing them into a complete polyhedral fan. We have a close look at the class of polynomials for which $\mathscr{il}(p)=\mathscr{cn}(p)$, whereas in general the quotient $\frac{\mathscr{il}(p)}{\mathscr{cn}(p)}$ is shown to be unbounded as the degree grows. Several examples and open questions are presented.

math.CO↗

arXiv2019

Convergence of multiple Fourier series and Pick's theorem

We add another brick to the large building comprising proofs of Pick's theorem. Although our proof is not the most elementary, it is short and reveals a connection between Pick's theorem and the pointwise convergence of multiple Fourier series of piecewise smooth functions.

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arXiv2019

Primitive points in rational polygons

Let $\mathcal A$ be a star-shaped polygon in the plane, with rational vertices, containing the origin. The number of primitive lattice points in the dilate $t\mathcal A$ is asymptotically $\frac6{π^2}$ Area$(t\mathcal A)$ as $t\to \infty$. We show that the error term is both $Ω_\pm\big( t\sqrt{\log\log t} \big)$ and $O(t(\log t)^{2/3}(\log\log t)^{4/3})$. Both bounds extend (to the above class of polygons) known results for the isosceles right triangle, which appear in the literature as bounds for the error term in the summatory function for Euler's $ϕ(n)$.

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arXiv2018

Fourier transforms of polytopes, solid angle sums, and discrete volume

Given a real closed polytope $P$, we first describe the Fourier transform of its indicator function by using iterations of Stokes' theorem. We then use the ensuing Fourier transform formulations, together with the Poisson summation formula, to give a new algorithm to count fractionally-weighted lattice points inside the one-parameter family of all real dilates of $P$. The combinatorics of the face poset of $P$ plays a central role in the description of the Fourier transform of $P$. We also obtain a closed form for the codimension-1 coefficient that appears in an expansion of this sum in powers of the real dilation parameter $t$. This closed form generalizes some known results about the Macdonald solid-angle polynomial, which is the analogous expression traditionally obtained by requiring that $t$ assumes only integer values. Although most of the present methodology applies to all real polytopes, a particularly nice application is to the study of all real dilates of integer (and rational) polytopes.

math.CO↗

arXiv2018

Cubic graphs, their Ehrhart quasi-polynomials, and a scissors congruence phenomenon

The scissors congruence conjecture for the unimodular group is an analogue of Hilbert's third problem, for the equidecomposability of polytopes. Liu and Osserman studied the Ehrhart quasi-polynomials of polytopes naturally associated to graphs whose vertices have degree one or three. In this paper, we prove the scissors congruence conjecture, posed by Haase and McAllister, for this class of polytopes. The key ingredient in the proofs is the nearest neighbor interchange on graphs and a naturally arising piecewise unimodular transformation.

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