arXiv subjects

Doyon Kim

Publications and source records attributed to Doyon Kim.

10 recordsLinked to original sources

arXiv2026

Quantum ergodicity of Eisenstein series for Bianchi groups

We prove the quantum ergodicity of Eisenstein series on the arithmetic hyperbolic 3-manifold $\operatorname{PSL}_2(\mathcal{O}_F)\backslash \mathbb{H}^3$, where $F$ is an imaginary quadratic field with ring of integers $\mathcal{O}_F$ and class number $h_F\geq 1$. This extends the work of Koyama, who proved the result in the case $h_F=1$, and establishes the first instance of quantum ergodicity of Eisenstein series over number fields with nontrivial class groups.

math.NT↗

arXiv2025

Component groups for non-supercuspidal $L$-parameters for $p$-adic $\mathrm{SL}_3$

We explicitly classify all the component groups associated to the non-supercuspidal, tempered $L$-parameters of $\mathrm{SL}_3(F)$ for a $p$-adic field $F$ of characteristic $0$ by direct case-by-case computations in $\mathrm{PGL}_3(\mathbb{C})$, following earlier work for $\mathrm{SL}_2$ by Labesse-Langlands and Shelstad.

math.RT↗

arXiv2025

Rational points on varieties defined by multihomogeneous diagonal forms

We give an asymptotic formula for the number of rational points of bounded height on algebraic varieties defined by systems of multihomogeneous diagonal equations. The proof uses the Hardy-Littlewood circle method and the hyperbola method developed by Blomer and Br\"udern.

math.NT↗

arXiv2024

Schubert cells and Whittaker functionals for $\text{GL}(n,\mathbb{R})$ part II: Existence via integration by parts

We give a new proof of the existence of Whittaker functionals for principal series representation of $\text{GL}(n,\mathbb{R})$, utilizing the analytic theory of distributions. We realize Whittaker functionals as equivariant distributions on $\text{GL}(n,\mathbb{R})$, whose restriction to the open Schubert cell is unique up to a constant. Using a birational map on the Schubert cells, we show that the unique distribution on the open Schubert cell extends to a distribution on the entire space $\text{GL}(n,\mathbb{R})$. This technique gives a proof of the analytic continuation of Jacquet integrals via integration by parts. We briefly discuss an application of the method to the Bessel functions on $\text{GL}(n,\mathbb{R})$.

math.RT↗

arXiv2024

Schubert cells and Whittaker functionals for $\text{GL}(n,\mathbb{R})$ part I: Combinatorics

We give a formula for a birational map on the Schubert cell associated to each Weyl group element of $G=\text{GL}(n)$. The map simplifies the UDL decomposition of matrices, providing structural insight into the Schubert cell decomposition of the flag variety $G/B$, where $B$ is a Borel subgroup. An application of the formula includes a new proof of the existence of Whittaker functionals for principal series representations of $\text{GL}(n,\mathbb{R})$ via integration by parts. In this paper, we establish combinatorial properties of the birational map and prove auxiliary results.

math.RT↗

arXiv2022

Infinitely many zeros of additively twisted $L$-functions on the critical line

For $f$ a cuspidal modular form for the group $Γ_0(N)$ of integral or half-integral weight, $N$ a multiple of $4$ in case the weight is half-integral, we study the zeros of the $L$-function attached to $f$ twisted by an additive character $e^{2πi n \frac{p}{q}}$ with $\frac{p}{q}\in \mathbb{Q}$. We prove that for certain $f$ and $\frac{p}{q}\in \mathbb{Q}$, the additively twisted $L$-function has infinitely many zeros on the critical line. We develop a variant of the Hardy-Littlewood method which uses distributions to prove the result.

math.NT↗

arXiv2017

On the Largest Integer that is not a Sum of Distinct Positive $n$th Powers

It is known that for an arbitrary positive integer $n$ the sequence $S(x^n)=(1^n, 2^n, \ldots)$ is complete, meaning that every sufficiently large integer is a sum of distinct $n$th powers of positive integers. We prove that every integer $m\geq (b-1)2^{n-1}(r+\frac{2}{3}(b-1)(2^{2n}-1)+2(b-2))^n-2a+ab$, where $a=n!2^{n^2}$, $b=2^{n^3}a^{n-1}$, $r=2^{n^2-n}a$, is a sum of distinct $n$th powers of positive integers.

math.NT↗

arXiv2016

Coloring the Real Line with Monochromatic Intervals

Let D be a finite set of positive real numbers. The distance graph G(R,D) is the graph with vertex set R (set of real numbers), and two vertices x, y are adjacent if |x-y| belongs to D. We prove that every positive integer t>1 there is a distance set D such that the chromatic number of G(R,D) is t and no proper coloring of G(R,D) with t colors allows monochromatic intervals. This result disproves a conjecture in [2].

math.CO↗

arXiv2016

Friends of 12

A friend of 12 is a positive integer different from 12 with the same abundancy index. By enlarging the supply of methods of Ward [1], it is shown that (i) if n is an odd friend of 12, then n=m^2, where m has at least 5 distinct prime factors, including 3, and (ii) if n is an even friend of 12 other than 234, then n=2*(q^e)*(m^2), in which q is a prime greater than or equal to 29, e is a positive integer, and both q and e are congruent to 1 mod 4, and m has at least 3 distinct odd prime factors, one of which is 3, and the other, none equal to q, are greater than or equal to 29.

math.HO↗

arXiv2016

2-Variable Frobenius Problem in Z[\sqrt M]

Suppose that m is a positive integer, not a perfect square. We present a formula solution to the 2-variable Frobenius problem in Z[\sqrt m] of the "first kind" ([3]).

math.NT↗