arXiv subjects

Junehyuk Jung

Publications and source records attributed to Junehyuk Jung.

25 records · Page 2Linked to original sources

arXiv2016

On tiling the integers with $4$-sets of the same gap sequence

Partitioning a set into similar, if not, identical, parts is a fundamental research topic in combinatorics. The question of partitioning the integers in various ways has been considered throughout history. Given a set $\{x_1, \ldots, x_n\}$ of integers where $x_1<\cdots<x_n$, let the {\it gap sequence} of this set be the nondecreasing sequence $d_1, \ldots, d_{n-1}$ where $\{d_1, \ldots, d_{n-1}\}$ equals $\{x_{i+1}-x_i:i\in\{1,\ldots, n-1\}\}$ as a multiset. This paper addresses the following question, which was explicitly asked by Nakamigawa: can the set of integers be partitioned into sets with the same gap sequence? The question is known to be true for any set where the gap sequence has length at most two. This paper provides evidence that the question is true when the gap sequence has length three. Namely, we prove that given positive integers $p$ and $q$, there is a positive integer $r_0$ such that for all $r\geq r_0$, the set of integers can be partitioned into $4$-sets with gap sequence $p, q$, $r$.

math.CO↗

arXiv2016

Quantitative quantum ergodicity and the nodal domains of Maass-Hecke cusp forms

We prove a quantitative statement of the quantum ergodicity for Hecke--Maass cusp forms on the modular surface. As an application of our result, along a density $1$ subsequence of even Hecke--Maass cusp forms, we obtain a sharp lower bound for the $L^2$-norm of the restriction to a fixed compact geodesic segment of $η=\{iy~:~y>0\} \subset \mathbb{H}$. We also obtain an upper bound of $O_ε\left(t_ϕ^{3/8+ε}\right)$ for the $L^\infty$ norm along a density $1$ subsequence of Hecke--Maass cusp forms; for such forms, this is an improvement over the upper bound of $O_ε\left(t_ϕ^{5/12+ε}\right)$ given by Iwaniec and Sarnak. In a recent work of Ghosh, Reznikov, and Sarnak, the authors proved for all even Hecke--Maass forms that the number of nodal domains, which intersect a geodesic segment of $η$, grows faster than $t_ϕ^{1/12-ε}$ for any $ε>0$, under the assumption that the Lindel{ö}f Hypothesis is true and that the geodesic segment is long enough. Upon removing a density zero subset of even Hecke--Maass forms, we prove without making any assumptions that the number of nodal domains grows faster than $t_ϕ^{1/8-ε}$ for any $ε>0$.

math.NT↗

arXiv2015

Quantum unique ergodicity and the number of nodal domains of eigenfunctions

We prove that the Hecke--Maass eigenforms for a compact arithmetic triangle group have a growing number of nodal domains as the eigenvalue tends to $+\infty$. More generally the same is proved for eigenfunctions on negatively curved surfaces that are even or odd with respect to a geodesic symmetry and for which Quantum Unique Ergodicity holds.

math.SP↗

arXiv2014

Number of nodal domains of eigenfunctions on non-positively curved surfaces with concave boundary

It is an open problem in general to prove that there exists a sequence of $Δ_g$-eigenfunctions $ϕ_{j_k}$ on a Riemannian manifold $(M, g)$ for which the number $N(ϕ_{j_k}) $ of nodal domains tends to infinity with the eigenvalue. Our main result is that $N(ϕ_{j_k}) \to \infty$ along a subsequence of eigenvalues of density $1$ if the $(M, g)$ is a non-positively curved surface with concave boundary, i.e. a generalized Sinai or Lorentz billiard. Unlike the recent closely related work of Ghosh-Reznikov-Sarnak and of the authors on the nodal domain counting problem, the surfaces need not have any symmetries.

math.SP↗

arXiv2013

Number of nodal domains and singular points of eigenfunctions of negatively curved surfaces with an isometric involution

We prove two types of nodal results for density one subsequences of an orthonormal basis $\{ϕ_j\}$ of eigenfunctions of the Laplacian on a negatively curved compact surface. The first type of result involves the intersections $Z_{ϕ_j} \cap H$ of the nodal set $Z_{ϕ_j}$ of $ϕ_j$ with a smooth curve $H$. Using recent results on quantum ergodic restriction theorems and prior results on periods of eigenfunctions over curves, we prove that under an asymmetry assumption on $H$, the number of intersection points tends to infinity for a density one subsequence of the $ϕ_j$. . We also prove that the number of zeros of the normal derivative $\partial_ν ϕ_j$ on $H$ tends to infinity. From these results we obtain a lower bound on the number of nodal domains of even and odd eigenfunctions on surfaces with an isometric involution. Using (and generalizing) a geometric argument of Ghosh-Reznikov-Sarnak, we show that the number of nodal domains of even or odd eigenfunctions tends to infinity for a density one subsequence of eigenfunctions.

math.SP↗

arXiv2012

On sparsity of positive-definite automorphic forms within a family

It is known due to Baker and Montgomery that almost all Fekete polynomials under certain ordering have at least one zero on the interval (0, 1). In terms of the positive-definiteness, Fekete polynomial has no zero on the interval (0, 1) if and only if the corresponding automorphic form is positive-definite. On generalizing their result, we formulate an axiomatic result about sets of automorphic forms satisfying certain averages when suitably ordered, which ensures that almost all forms are not positive-definite within such sets. We then apply the result to various families, including the family of holomorphic cusp forms, the family of the Hilbert class characters of imaginary quadratic fields, and the family of elliptic curves.

math.NT↗

arXiv2011

Pretentiously detecting power cancellation

Granville and Soundararajan have recently introduced the notion of pretentiousness in the study of multiplicative functions of modulus bounded by 1, essentially the idea that two functions which are similar in a precise sense should exhibit similar behavior. It turns out, somewhat surprisingly, that this does not directly extend to detecting power cancellation - there are multiplicative functions which exhibit as much cancellation as possible in their partial sums that, modified slightly, give rise to functions which exhibit almost as little as possible. We develop two new notions of pretentiousness under which power cancellation can be detected, one of which applies to a much broader class of multiplicative functions.

math.NT↗