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Henry Shin

Publications and source records attributed to Henry Shin.

11 recordsLinked to original sources
arXiv2026

Iterated-sumset spectra: The complete exponent law and its rank geometry

For integers $h,k\geq 1$, let $hA$ be the $h$-fold sumset of $A$ and put $\mathcal{R}(h,k)=\{|hA|:A\subset\mathbb{Z}, |A|=k\}$. Previously, the fixed-cardinality exponent law was known only for $k\leq 3$; every fixed $k\geq 4$ remained open. We settle the problem in full by determining the complete fixed-cardinality exponent law: $|\mathcal{R}(h,k)|=\begin{cases}1,&k\leq 2,\\ h,&k=3,\\ h^{k-1+o_k(1)},&k\geq 4\end{cases}$. Here $o_k(1)\to 0$ as $h\to\infty$ with $k$ fixed. More sharply, for fixed $k\geq 4$, an interval of length $\Theta_k(h^{k-1})$ contains at least $h^{k-1-o_k(1)}$ attainable values. At $k=4$ we prove $|\mathcal{R}(h,4)|=\Theta(h^3)$ with positive lower density in its ambient interval, disproving Nathanson's proposed $o(h^3)$ and $O(h^2)$ bounds. One bounded addition-table geometry drives these results, coupling Hilbert-energy amplification to optimal finite-observation compression. Every ordered real $k$-set ($k\geq 2$) has an integer model in $[0,O_k(h^{k-2})]$ preserving every sum equality and strict comparison through degree $h$; the exponent $k-2$ is sharp. The universal label-realization length is therefore $\Theta_k(h^{k-2})$, one power sharper than Nathanson's $O_k(h^{k-1})$ bound. For $h\geq 2$ and $k\geq 3$, minimum active rank equals realization-frequency codimension, exponent-shape codimension, and sampling-rarity exponent; a full-exponent family has maximal-rank witnesses with Cohen-Macaulay toric coordinate rings. At rank zero, for $h\geq 2$, it proves the conjectural OEIS A227589 formula $\binom{h+2}{2}+\mathbf{1}_{\{2\nmid h\}}$ for the least normalized diameter of a four-point $B_h$-set. It also gives exact fixed-$(h,k)$ popularity laws for $k$-subsets of $\{1,\ldots,q\}$ as $q\to\infty$, resolving Nathanson's Problems 9 and 10.

math.CO↗
arXiv2026

Open full-metric formation and local marked first-order asymptotic moduli of FIK blowdown singularities

We prove that the Feldman--Ilmanen--Knopf (FIK) blowdown singularity has a nonempty open nonlinear formation basin in the full space of smooth Riemannian metrics and, on a smaller neighborhood, local marked first-order asymptotic moduli. Given a closed connected oriented Riemannian four-manifold, an implantation point, and a positive time bound, its oriented blow-up admits a relatively $C^{2,\alpha}$-open set of smooth metrics whose Ricci flows form, before that time, a localized FIK singularity with global Type-I curvature control. No symmetry, K\"ahler condition, or finite-dimensional tuning is imposed. Curvature stays uniformly bounded outside the implantation region; fixed-convention marked rescalings converge along the full singular-time sequence to the ancient FIK flow; and the exceptional sphere has sharp FIK asymptotics for area, intrinsic diameter, and curvature. To our knowledge, this is the first open full-metric formation theorem for a finite-time singularity modeled on a noncompact, noncylindrical shrinker. A self-contained spectral certification identifies the weighted FIK operator's nine-dimensional nonnegative space with the scaling and diffeomorphism directions, and exact modulation removes this block. On a smaller little-H\"older $h^{2,\alpha}$ neighborhood, a fixed positive-time restart yields the marked first-profile coordinate $\mathfrak{A}_1=\lambda_\infty^{-\gamma_1}V_\infty\in E_1$. This amplitude is a split $C^1$ submersion and locally the projection onto $E_1$. Equality of amplitudes characterizes marked first-order agreement; a transverse disk uniquely realizes each sufficiently small amplitude; normalized two-flow differences converge to the corresponding Jacobi field; and the amplitude determines the first quadratic scale and phase response. This gives a complete local marked first-order profile coordinate for FIK blowdown.

math.DG↗
arXiv2026

A global girth obstruction for Garg--Mineyev taiko product structures

Mineyev's taiko construction, in Garg--Mineyev's finite support-size formulation, gives a concrete route from finite support data to zero divisors and units in group rings of torsion-free CAT(0) groups over $\mathbb{F}_2$. We prove that this triple-girth product-structure route is globally closed: no product structure, even or odd, with support sizes $m,n\ge2$ admits a coherent orientation for which the no-fold and triple-girth conditions both hold. Consequently the Garg--Mineyev triple-girth product-structure assembly route produces neither zero-divisor nor unit counterexamples over $\mathbb{F}_2$ for any such support-size pair. The obstruction is structural, not a bounded-search artifact. High middle-link girth forces signed colors into a balanced near-disjoint rectangle decomposition of the board, with the single odd defect omitted. The product identity, pressure inequalities, Fisher inequalities, and a dual Fisher bound force the middle link to have girth $4$ or $6$; in the girth-six case, the minimum of the two horizontal-link girths is at most $5$. This dichotomy rules out every triple-girth branch. A weighted dual Fisher inequality and an exact finite certificate sharpen the frontier: if the middle link has girth $6$, the horizontal girth is at most $4$, and characteristic-two affine-plane constructions attain equality. Thus the Garg--Mineyev finite failures reflect a structural barrier in the taiko geometry itself. The finite certificate is used only for this sharper frontier, not for the no-$T_4$ obstruction.

math.CO↗
arXiv2026

From rough local mass to logarithmic spectral ladders for damped waves

We prove an exact inverse theorem for damped waves on $X=\mathbb{T}\times Y$ with arbitrary bounded measurable transverse damping $a=a(x)\geq 0$. Let $\Theta_a(r)$ be the least average of $a$ on radius-$r$ intervals. Then $\liminf_{r\downarrow 0}\Theta_a(r)>0$ characterizes exponential stability, while for every $L(S)=\ell(\log(eS))$ with $\ell:[1,\infty)\to[1,\infty)$ nondecreasing, unbounded, and eventually doubling, $\Theta_a(r)\gtrsim L(1/r)^{-1}$ is equivalent to stationary- and generator-resolvent bounds $O(|s|^{-1}L(|s|))$ and $O(L(|s|))$. One resonant scalar block per dyadic octave already recovers this multiscale mass bound. Every such critical profile occurs for indicator damping on an open dense set of arbitrarily small measure, whereas the stationary resolvent-to-mass implication fails at every sublinear power gauge. For the cusp $a(x)=(\log(e/|x|))^{-A}$, $A>0$, near its isolated damping well, we prove the sharp obstruction is genuine spectrum. Blow-up yields $-\partial_y^2+i\log|y|$, whose spectrum is a simple interlaced vertical ladder. Every fixed finite ladder portion transfers to damped-wave eigenvalues with complete blockwise algebraic count and two-term asymptotics. Combined with the resolvent theorem, these eigenvalues give the sharp regularized decay scale $\exp[-t^{1/(A+1)}]$.

math.AP↗
arXiv2026

Exchange identities and symmetric slices of the valley Delta conjecture

The valley Delta conjecture of Haglund, Remmel and Wilson predicts that the symmetric function $\Delta'_{e_{n-k-1}} e_n$ equals a generating function over labelled Dyck paths with $k$ decorated contractible valleys. Unlike the rise version, which is now a theorem, the valley version remains open; indeed it is not even known that its combinatorial side is symmetric. We prove that the coefficients of $t^0$ and $t^1$ in the valley generating function are symmetric functions for all $n\ge 1$ and $k\ge 0$; equivalently, the fixed-diagonal-multiset slices of area at most one are symmetric. The theorem follows from an adjacent exchange identity for scaffold classes, which we prove in a strictly stronger form refined by the numbers of undecorated rows carrying the labels $r,r+1$ between consecutive rows with other labels. The proof develops a transfer-operator calculus in a $q$-deformed two-variable algebra generated by two commuting half-twists. In this algebra the exchange reduces to two scalar symmetric-series identities for the operator $T=u\mathfrak{d}+v$. We also verify the refined identity computationally at area two over extensive finite ranges and state the resulting general conjecture.

math.CO↗
arXiv2026

One extra edge forces Berge pancyclicity

We resolve a question of Bailey, Hollars, Li and Luo. For all sufficiently large $n$, let $r=\lfloor(n-1)/2\rfloor$. We prove that the edges of any Hamiltonian Berge cycle in a simple $n$-vertex $r$-uniform hypergraph, together with any one additional edge, contain Berge cycles of every length from $2$ to $n$. In odd order we prove a stronger prescribed-unused-edge theorem using rigidity of large subsets of odd cyclic groups and an alternating matching exchange. In even order we introduce a two-gap edge-reassignment method. Split locks cover all lengths outside a seven-term middle band. The absence of the central length forces an exact reflected translation-wave structure, which is eliminated by an additive covering theorem derived from sum-free stability. The remaining near-central lengths follow from a two-defect recurrence and bounded-run forcing.

math.CO↗
arXiv2026

Hilbert-90 quotient maps, torsion defects, and symmetric monodromy

Let $\tau(z)=-1-z^{-1}$. We study the reduced rational maps $h_d:\mathbb{P}^1\to\mathbb{P}^1$ obtained by cancelling common factors in $H_d^{\rm raw}(z)=z^d(\tau(z)^d-1)/(z^d-1)$. These maps arise by Hilbert-90 descent from the trace-zero maps $X^{dq}-X^d$ on $\ker\operatorname{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}$, but the principal object is the resulting $\tau$-equivariant quotient-map family; nonconstant separable members are viewed as covers. We prove that cancellation is exactly a torsion-defect phenomenon. If $\ell(-)$ denotes scheme-theoretic length and $\boldsymbol{\mu}_d=\ker([d]:\mathbb{G}_m\to\mathbb{G}_m)$, then $\mathrm{deg}(h_d)=d-\ell((1+X+Y=0)\cap\boldsymbol{\mu}_d^2)$, and, in characteristic $p>0$ with $d=p^s d_0$ and $p\nmid d_0$, $h_d=\operatorname{Frob}_{p^s}\circ h_{d_0}$ and $\mathrm{deg}(h_d)=p^s\mathrm{deg}(h_{d_0})$. We classify the tame quotient strata of morphism degree at most one and exactly two; the maximal-defect stratum yields a characteristic-two Mersenne trace-zero permutation family. In characteristic zero we prove the main monodromy theorem: every non-linear quotient is Morse and has full symmetric geometric monodromy, $G_{h_d}=S_{\mathrm{deg}(h_d)}$; the proof rules out branch-value collisions via a cyclotomic cross-ratio equation. In positive characteristic we isolate Frobenius-sparse Kummer and Artin-Schreier quotients, a certificate-verified characteristic-19 Klein-four Galois quotient, and the first nonsparse Frobenius-lacunary tower up to its stated primitivity and wild-inertia boundary. A twisted off-diagonal fiber-square trace formula turns $2$-transitive monodromy into a uniform obstruction to $\tau$-twisted exceptionality.

math.NT↗
arXiv2026

Coset-refined trace statistics, nodal characters, and affine branches in cubic norm tori

Prescribed trace/norm estimates and Soto-Andrade-type sums control whole fibers or related global character sums. We prove a coset-refined trace theorem for cubic norm-one tori. Let $B/\mathbb{F}_q$ be finite \'etale cubic, $\operatorname{char}\mathbb{F}_q\ne2,3$, and let $T_B=\ker(\operatorname{N}_{B/\mathbb{F}_q}:\operatorname{Res}_{B/\mathbb{F}_q}\mathbb{G}_m\to\mathbb{G}_m)$. For every subgroup $H\subset T_B(\mathbb{F}_q)$ of index $m$, every coset $gH$, every $\gamma\in B^\times$, and every smooth fiber $\operatorname{Tr}(\gamma h)=s$, $s^3\ne27\operatorname{N}(\gamma)$, we prove $N_{gH,B}(s;\gamma)=m^{-1}N_B(s,\operatorname{N}\gamma)+E_{gH,B}(s;\gamma)$, with $|E_{gH,B}(s;\gamma)|\le3(1-1/m)\sqrt q$. The geometric input is a Picard-Kummer kernel calculation: no nontrivial torus character becomes geometrically constant on a smooth trace/norm curve, so nontrivial coset character sums have square-root cancellation. On the nodal boundary $s^3=27\operatorname{N}(\gamma)$, the kernel degenerates exactly to a cyclic cubic Kummer kernel. Its Frobenius-fixed part is the sole source of order-$q$ bias; after removing that explicit projection, remaining characters again have square-root cancellation up to bounded normalization/node correction. The same geometry gives local branch theory for $\operatorname{Tr}_A(\gamma\eta^n)=c$ over finite \'etale cubic $\mathbb{Z}_p$-algebras, $p\ge5$. The logarithmic tangent and trace-dual codifferent coordinates identify singular branches: nondegenerate classes have quadratic Hensel models, while the genuinely affine degenerate class has a cubic first-obstruction model; in full norm-fiber orbits singular branch counting reduces to one cubic norm equation.

math.NT↗
arXiv2026

Logarithmic source curves in polynomial fiber products

Let $k$ be a field of characteristic zero and let $f,g\in k[x]$ be nonconstant. We study rational lifts of $f(a)$ through $g$ that do not arise from a composition $f=g\circ h$. To each non-graph component of $f(X)=g(Y)$ we attach its logarithmic source curve, namely the smooth compactification of its normalization with reduced boundary. The main geometric result is a sharp contact formula at infinity: if $N=\deg g/\gcd(\deg f,\deg g)$, then every one-infinity non-graph source has $X$-degree $N$, and in general the $X$-degree is $N$ times the number of boundary points. Over number fields this yields a finite symmetric-difference expansion of $S$-integral new lifts. Active one-infinity sources give exactly the power terms in height counts; positive-rank admissible two-infinity sources give logarithmic $S$-unit families; and inactive one-infinity sources, rank-zero two-infinity sources, and the remaining components contribute only finitely many inputs. Primitive one-infinity source classes have only polylogarithmic overlap, and ordered configuration covers introduce no new exponent. Over $\mathbb Q$, the $B^{1/2}$ boundary is precisely the quadratic Bilu--Tichy source cell.

math.NT↗
arXiv2025

Correction to: Curvature growth of some 4-dimensional gradient Ricci soliton singularity models

This note corrects an error in the proof of Proposition 13 in arXiv:1903.09181 and simultaneously establishes a more general result. We prove that if $M $ is a compact connected oriented $4$-manifold with connected boundary $\partial M$, and if an unbounded number of disjoint copies of $M$ embed topologically and locally flatly in the interior of a compact $4$-manifold $N,$ then $\operatorname{Tor}H_1(\partial M;\mathbb{Z})$ is a direct double, i.e., $\operatorname{Tor}H_1(\partial M;\mathbb{Z})\cong A \oplus A$, with the linking pairing vanishing identically on the first summand, i.e., the linking pairing is split metabolic. This partially generalizes Hantzsche's theorem stating that the linking pairing for a closed $3$-manifold that embeds in $S^4$ is hyperbolic.

math.DG↗