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Sanghoon Kwon

Publications and source records attributed to Sanghoon Kwon.

At least 19 recordsLinked to original sources
arXiv2026

Zeta renormalization and pressure at infinity for an infinitely cusped tree lattice

We study weighted periodic-orbit zeta functions for an infinitely cusped tree lattice $Y_q$, where $q\ge2$ is even and the quotient is a one-sided comb. The global Euler product fails coefficientwise because infinitely many primitive cycles have length four. A first-return determinant at a finite directed-edge set nevertheless exists, and stationary Schur elimination gives an algebraic formula for the root local zeta and its dominant poles. For the two-step multiplicity potential we compute the Gurevich pressure $P_G=2\log(q+1)$ and pressure at infinity $P_\infty=\log(4q)$, yielding strong positive recurrence and exponential local-orbit asymptotics. The height-damped transition operator is trace class. After subtraction of an explicit integrated-pressure counterterm, the inverse Fredholm determinant has a locally uniform finite part, expressed by a convergent dilogarithmic product and covariant under changes of height.

math.DS↗
arXiv2026

Exact renewal laws for minimal common-denominator profiles in simultaneous Laurent-series approximation

Let $\alpha_1,\ldots,\alpha_r$ be independent Haar-random fractional Laurent series over $\mathbb{F}_q$, and let $L_r(n)$ be the least coefficient length of a polynomial denominator that simultaneously cancels the first $n$ negative coefficients. We prove that the minimal kernel is a line and that the residual vectors revealed immediately after the stopping times $T_n=n+L_r(n)-1$ are iid uniform on $\mathbb{F}_q^r$. Hence the jump indicators of $L_r(n)$ are iid Bernoulli variables with parameter $1-q^{-r}$; conditionally on a jump, the residual direction is uniform on $\mathbb P^{r-1}(\mathbb{F}_q)$. We also give an exact kernel-growth clock for positive jump sizes and a geometric tail bound uniform in the depth; for two series the jump is decided at the first or second kernel-growth epoch with probabilities $q^{-1}$ and $1-q^{-1}$. The marked renewal law yields exact binomial and fluctuation laws in the depth variable and the density of newly attained minimal denominator lengths $\frac{1-q^{-r}}{r}$ in the coefficient-length variable. For $r=1$ this is the classical iid partial-quotient degree law in the depth coordinate, for which we give an exact dictionary. The new probabilistic content is the simultaneous common-denominator law for $r\ge2$. We also establish exact profile-correspondence and record-duality formulas with joint linear complexity.

math.PR↗
arXiv2026

Vertex volumes, lattice-minima tails, and height zeta functions for the standard arithmetic quotient of $\operatorname{PGL}_d$

We study the standard nonuniform arithmetic quotient of the affine Bruhat--Tits building attached to $\operatorname{PGL}_d(\mathbb F_q(\!(t^{-1})\!))$, with Haar measure normalized so that a maximal compact subgroup has volume one. We first compute its vertex volume in closed product form. The proof is entirely building-theoretic: vertices are parametrized by a dominant sector, their stabilizers are counted exactly, and the resulting sum over block compositions is evaluated by a cut-set recursion. On the same quotient, we introduce a homothety-invariant normalized lattice-minima height $\alpha$. We determine its exact integrability threshold, proving that $\alpha$ belongs to $L^r$ precisely for $0<r<d$, and establish a sharp cusp-tail estimate of order $T^{-d}$. The associated positive-moment height zeta function, equivalently the Mellin transform of the cusp-height distribution, converges exactly in the half-plane $\operatorname{Re}(s)<d$. It admits a meromorphic continuation as a rational function of $q^{s/d}$ and has a simple pole at $s=d$, with an explicit critical coefficient. We also compute the resulting rational functions explicitly for $d=3,4,5$. Thus the same dominant-sector coordinates simultaneously control volume, cusp decay, and the analytic structure of the height zeta function.

math.NT↗
arXiv2026

Critical-exponent spectra and rank two inverse realization on biregular trees

We study the critical-exponent, or equivalently entropy, spectrum arising from free type-preserving actions on the biregular tree $\mathcal T_{r+1,s+1}$. For an action with a nonempty finite quotient core, the critical exponent agrees with both the volume entropy of the universal cover of the core and the topological entropy of the associated non-backtracking edge shift. The unrestricted spectrum is $[0,\frac12\log(rs)]$, whereas the finitely generated spectrum is a countable dense subset of this interval obtained by taking logarithms of Hashimoto spectral radii. We stratify this finite-state spectrum by the circuit rank of the quotient core. At each fixed rank, finitely many typed kernels parametrize all values. Moreover, every positive entropy value has only finitely many kernel--length realizations, up to type-preserving isomorphism, and every positive accumulation point of a fixed-rank spectrum belongs to a lower-rank stratum. At rank two, the corresponding exponential rates admit a complete inverse classification in terms of three explicit polynomial families, together with a finite exact membership test for algebraic-integer inputs.

math.GR↗
arXiv2026

$K$-spherical horospherical averages on the Nagao quotient: tree combinatorics and exact discrepancy

Let \[ F=\mathbb{F}_q(\!(t^{-1})\!),\qquad G=\mathrm{SL}_2(F),\qquad \Gamma=\mathrm{SL}_2(\mathbb{F}_q[t]),\qquad X=\Gamma\backslash G, \] and let $K=\mathrm{SL}_2(\mathcal{O})$, where $\mathcal{O}=\mathbb{F}_q[\![t^{-1}]\!]$. We study right $K$-spherical averages along the upper unipotent subgroup, the horospherical subgroup associated with the standard cusp, on the Nagao lattice quotient. The basic observation is that the $K$-spherical projection converts two natural dynamical families - expanding translates of compact unipotent orbits and cusp-adapted truncations of dense unipotent orbits - into the same rooted descendant problem on the Bruhat--Tits tree. In the even bipartite sector the limiting height law is the explicit probability measure \[ \rho^{\mathrm{ev}}(0)=\frac{q-1}{q},\qquad \rho^{\mathrm{ev}}(2m)=(q^2-1)q^{-2m-1}\qquad (m\ge 1). \] We prove an exact discrepancy formula: in the backward state the error is a pure top-shell term minus a missing tail, while in the forward state the error is a first-turn weighted sum of backward errors. These formulas give quantitative $K$-spherical equidistribution for expanding translates of compact $U$-orbits and for dense-orbit truncations. For compactly supported $K$-spherical observables in the expanding translates of compact orbits, the discrepancy is eventually exactly zero. In the dense case the rate is controlled by the continued-fraction expansion of the boundary point attached to the orbit.

math.DS↗
arXiv2026

Special periods and some non-tempered cases of the Gan-Gross-Prasad conjecture

In this paper, we establish a relationship between special periods and special L-values of automorphic representations of classical groups, and prove the non-tempered global Gan--Gross--Prasad conjecture in several cases. Our approach consists of two main steps. First, inspired by Rallis' tower property, we study the interaction between special periods and the tower property for the genericity of global theta lifts. Second, we investigate the relationship between the analytic properties of L-functions and special periods via the Rankin--Selberg integral method. Combining these results with non-vanishing criteria for global theta lifts in terms of various L-values, we prove three explicit higher-corank families of non-tempered cases of the global Gan--Gross--Prasad conjecture.

math.NT↗
arXiv2025

Chamber zeta function and closed galleries in the standard non-uniform complex from $\operatorname{PGL}_3$

We introduce the \emph{chamber zeta function} for a complex of groups, defined via an Euler product over primitive tailless chamber galleries, extending the Ihara--Bass framework from weighted graphs to higher-rank settings. Let $\mathcal{B}$ be the Bruhat--Tits building of $\mathrm{PGL}_{3}(F)$ for a non-archimedean local field $F$ with residue field $\mathbb{F}_{q}$. For the standard arithmetic quotient $\Gamma\backslash\mathcal{B}$ with $\Gamma=\mathrm{PGL}_{3}(\mathbb{F}_{q}[t])$, we prove an Ihara--Bass type \emph{determinant formula} expressing the chamber zeta function as the reciprocal of a characteristic polynomial of a naturally defined chamber transfer operator. In particular, the chamber zeta function is \emph{rational} in its complex parameter. As an application of the determinant formula, we obtain explicit counting results for closed gallery classes arising from tailless galleries in $\mathcal{B}$, including exact identities and spectral asymptotics governed by the chamber operator.

math.NT↗
arXiv2024

Edge zeta function and closed cycles in the standard non-uniform complex from $\operatorname{PGL}_3$

In this paper, we define the edge zeta function of weighted complex. We also present the formula for the edge zeta function of the standard non-uniform complex $\operatorname{PGL}(3,\mathbb{F}_q[t])\backslash\operatorname{PGL}(3,\mathbb{F}_q(\!(t^{-1})\!))/\operatorname{PGL}(3,\mathbb{F}_q[\![t^{-1}]\!])$, arising from the group $\operatorname{PGL}_3$, as a rational function. Applying trunction in a specific direction is one of the main ingredient. As a result, we obtain the exact formula for the number of closed cycles coming from geodesics in the building.

math.GR↗
arXiv2024

The tower property on the genericity of global theta lifts

In this paper, we examine the tower property concerning the genericity of global theta lifts between various classical groups, drawing inspiration from Rallis' tower property. By exploring the relationship between the analytic properties of $L$-functions and special Bessel and Fourier-Jacobi periods, we demonstrate that the first occurrence of global theta lifts between dual reductive groups preserves genericity. As an application, we establish the global Gan-Gross-Prasad conjecture for $\SO_{2n+1} \times \SO_{2}$ under the assumption that $\SO_{2}$ is split and its representation is trivial.

math.NT↗
arXiv2023

Weak Ramanujan property of the standard non-uniform arithmetic quotient of $PGL_4$

Let $F$ be a field of formal series over a finite field and $\mathcal{B}_d$ be the affine building associated to $PGL_d(F)$. Given a lattice $Γ$ in $PGL_d(F)$, the complex arising as a quotient $Γ\backslash \mathcal{B}_d$ is called weakly Ramanujan if every non-tivial discrete simultaneous spectrum of the colored adjacency operators $A_1,A_2,\ldots,A_{d-1}$ acting on $L^2(Γ\backslash \mathcal{B}_d)$ is contained in the simultaneous spectrum of those operators acting on $L^2(\mathcal{B})$. In this paper, we prove that the standard non-uniform arithmetic quotient $PGL_4(\mathbb{F}_q[t])\backslash \mathcal{B}_4$ of $PGL_4(F)$ is weakly Ramanujan.

math.NT↗
arXiv2023

Zeta functions of geometrically finite graphs of groups

In this paper, we explore the properties of zeta functions associated with infinite graphs of groups that arise as quotients of cuspidal tree-lattices, including all non-uniform arithmetic quotients of the tree of rank one Lie groups over local fields. Through various examples, we illustrate pairs of non-isomorphic cuspidal tree-lattices with the same Ihara zeta function. Additionally, we analyze the spectral behavior of a sequence of graphs of groups whose pole-free regions of zeta functions converge towards 0, which also presents an example of arbitrary small exponential error-term in counting geodesic formula.

math.GR↗
arXiv2023

The local converse theorem for quasi-split $O_{2n}$ and $SO_{2n}$

Let $F$ be a non-archimedean local field of characteristic not equal to 2. In this paper, we prove the local converse theorem for quasi-split $\O_{2n}(F)$ and $\SO_{2n}(F)$, via the description of the local theta correspondence between $\O_{2n}(F)$ and $\Sp_{2n}(F)$. More precisely, as a main step, we explicitly describe the precise behavior of the $\gamma$-factors under the correspondence. Furthermore, we apply our results to prove the weak rigidity theorems for irreducible generic cuspidal automorphic representations of $\O_{2n}(\A)$ and $\SO_{2n}(\mathbb{A})$, respectively, where $\A$ is a ring of adele of a global number field $L$.

math.NT↗
arXiv2021

Spectrum of weighted adjacency operator on a non-uniform arithmetic quotient of $PGL_3$

We investigate the automorphic spectra of the natural weighted adjacency operator on the complex arising as a $PGL(3,\mathbb{F}_q[t])$ quotient of $\widetilde{A}_2$-type building. We prove that the set of non-trivial approximate eigenvalues $(λ^+,λ^-)$ of the weighted adjacency operators $A_w^\pm$ on the quotient induced from the colored adjacency operators $A^\pm$ on the building for $PGL_3$ contains the simultaneous spectrum of $A^\pm$ and another hypocycloid with three cusps. As a byproduct, we re-establish a proof of the fact that $PGL(3,\mathbb{F}_q[t])\backslash PGL(3,\mathbb{F}_q(\!(t^{-1})\!))/PGL(3,\mathbb{F}_q[\![t^{-1}]\!])$ is not a Ramanujan complex, from a combinatorial aspect.

math.NT↗
arXiv2019

A combinatorial approach to the Littlewood conjecture in a field of formal series

A long-standing conjecture of Littlewood about simultaneous Diophantine approximation has an analogous problem for a field of formal Laurent series $\mathbb{F}(\!(t^{-1})\!)$. That is, we can ask whether for any series $Θ$, $Φ$ and any $ε>0$, there is a polynomial $α$ such that $|α|\langleαΘ\rangle\langleαΦ\rangle$ where $\langleΘ\rangle=\underset{β\in\mathbb{F}[t]}{\inf}|Θ-β|$. If the base field $\mathbb{F}$ is infinite, then the answer is negative due to Davenport and Lewis (1963). We give a connection between the combinatorics of an orbit under a semigroup action and Diophantine approximation problem when $\mathbb{F}$ is finite.

math.NT↗
arXiv2019

Limiting distribution of geodesics in a geometrically finite quotients of regular trees

In this article, we prove an extreme value theorem on the limit distribution of geodesics in a geometrically finite quotient of $Γ\backslash\mathcal{T}$ a locally finite tree. Main examples of such graphs are quotients of a Bruhat-Tits tree $\mathcal{T}$ by non-cocompact discrete subgroups $Γ$ of $PGL(2,\mathbf{K})$ of a positive characteristic local field $\mathbf{K}$. We investigate, for a given time $T$, the measure of the set of $Γ$-equivalent geodesic classes which stay up to time $T$ the region of distance $d$ at most $N$ depending on $T$ from a fixed compact subset $D$ of $Γ\backslash\mathcal{T}$. Namely, for Bowen-Margulis measure $μ$ on the space $Γ\backslash\mathcal{GT}$ of geodesics and the critical exponent $δ$ of $Γ$, we show that there exists a constant $C$ depending on $Γ$ and $D$ such that $$\lim_{T\to\infty}μ\left(\left\{[l]\inΓ\backslash\mathcal{GT}\colon \underset{0\le t \le T}{\textrm{max}}d(D,l(t))\le N+y\right\}\right)=e^{-q^y/e^{2δy}}$$ with $$N=\log_{e^{2δ/q}}\left(\frac{T(e^{2δ-q)}}{2e^{2δ}-C(e^{2δ}-q)}\right).$$

math.DS↗
arXiv2018

On the set of critical exponents of discrete groups acting on regular trees

We study the set of critical exponents of discrete groups acting on regular trees. We prove that for every real number $δ$ between $0$ and $\frac{1}{2}\log q$, there is a discrete subgroup $Γ$ acting without inversion on a $(q+1)$-regular tree whose critical exponent is equal to $δ$. Explicit construction of edge-indexed graphs corresponding to a quotient graph of groups are given.

math.DS↗