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Scott Duke Kominers

Publications and source records attributed to Scott Duke Kominers.

At least 19 recordsLinked to original sources
arXiv2026

A proper Euler magic matrix of order $5$

An Euler magic matrix is an integer matrix $M$ with $MM^{t}=γI$ whose squared entries sum to $γ$ along both main diagonals; it is proper if its squared entries are pairwise distinct. Euler constructed an order-$4$ proper example, and Müller settled orders $3$ (none exist) and $8$, leaving order $5$ as the smallest open case. We construct such a matrix, by rotating one of Müller's "near-misses" under a mirror-symmetric coordinate pair so that the two diagonal conditions collapse to a single rational equation; the same invariant suggests a uniform approach to the odd orders.

math.GM↗
arXiv2026

Long Intervals Without Distinct Multiples of the First $n$ Positive Integers

For positive integers $n$ and $m$, let $f(n,m)$ be the least integer $h\ge0$ such that $(m,m+h]$ contains distinct integers $a_1,\ldots,a_n$ satisfying $i\mid a_i$ for $1\le i\le n$, and put $F(n)=\max_{m\in\mathbb{N}} f(n,m)$. A recent theorem of van Doorn [INTEGERS, 2026; arXiv:2601.16972] gives $F(n)-f(n,n)>0.36\,n\log n/\log\log n$ for sufficiently large $n$. We prove \[ \liminf_{n\to\infty} \frac{F(n)-f(n,n)}{n\log n} \ge \frac{1}{\mathrm{e}}. \] Thus, for every fixed $c<1/\mathrm{e}$ and all sufficiently large $n$, some interval of length $c\,n\log n$ contains no system of pairwise distinct multiples of $1,2,\ldots,n$. The proof applies an Erdős--Pomerance smooth-number obstruction at starting points $m\asymp n\log n$, using local saddle-point estimates of Hildebrand and Tenenbaum.

math.NT↗
arXiv2026

Palindromes on the $τ$-circle: A note for Palindrome Tau Day, 6/28/26

An integer palindrome is a self-reciprocal polynomial evaluated at its base, so its roots are symmetric about the unit circle -- where the coordinate is angle, in turns of $τ$. Read this way, the date $\texttt{6/28/26}\to 62826$ secretly contains the primitive cube roots of unity -- at angle $τ/3$ -- along with one further pair of roots on the circle.

math.HO↗
arXiv2026

For which real quadratic fields is Kim's octonary form universal?

Let $K=\mathbb{Q}(\sqrt{D})$ with $D>1$ squarefree, and let $\varepsilon_+$ be the totally positive fundamental unit of $\mathcal{O}_K$. B. M. Kim proved in 2000 that the octonary diagonal form \[ f=x_1^2+\cdots+x_4^2+\varepsilon_+(x_5^2+\cdots+x_8^2) \] is universal over $\mathcal{O}_K$ whenever $D=n^2-1$ is squarefree. We complete Kim's result to an if-and-only-if classification: $f$ is universal if and only if $D=n^2-1$ for some $n\ge2$, or $D=n^2-4$ for some odd $n\ge3$, in both cases subject to squarefreeness. The second family appears to be new in this context and contains $\mathbb{Q}(\sqrt{5})$ at $n=3$ as a degenerate boundary case, recovering Maaß's classical exceptional three-square phenomenon. Equivalently, $f$ is universal over $\mathcal{O}_K$ if and only if the Blomer--Kala invariant $M_D$ equals $1$; for the two stated families we have $M_D=1$, so the Blomer--Kala universal $8M_D$-variable construction specializes exactly to $f$. The converse leverages a continued-fraction involution $τ(γ)=\varepsilon_+γ'$ together with a closed formula in convergent coordinates, a three-consecutive-square-values lemma for primitive quadratic polynomials of positive fundamental discriminant, and an even-root exclusion lemma derived from complete-quotient norm bounds.

math.NT↗
arXiv2026

A reduced planar body with area greater than $πΔ^2/4$

We construct a reduced planar convex body $R$ with thickness $Δ(R)=1$ and \[\operatorname{area}(R)=0.786215\ldots>0.785398\ldots=\fracπ{4}.\] Thus $R$ is a counterexample to Lassak's conjectured upper bound $\operatorname{area}\le(π/4)Δ^2$ for planar reduced bodies. The construction is given by an explicit support function, and the proofs use only elementary support-function, width, area, and contact-point computations.

math.MG↗
arXiv2026

An eigenvalue proof of Hegedüs's bound for codes with a single Hamming distance

We give a short, self-contained linear-algebra proof of a bound of Hegedüs [Australasian Journal of Combinatorics, 2026; arXiv:2409.07877]: if all pairwise Hamming distances in a family of subsets of $\{1,\ldots,n\}$ equal a fixed value $λ\ne(n+1)/2$, then the family has at most $n$ members. Our proof uses the same Gram matrix as in Hegedüs's argument, but reads its eigenvalues in place of its determinant, and keys off of a single fact about vectors of equal norm and equal pairwise inner product. That fact applies verbatim over an alphabet of size $q$, where it yields the bound $n(q-1)$ for $λ\ne\bigl((q-1)n+1\bigr)/q$ -- the corrected form of a conjecture of Hegedüs, recently established by Hu, Huang, and Yu [arXiv:2504.07036].

math.CO↗
arXiv2026

On the degree-$2$ Siegel theta series of extremal even unimodular lattices of ranks $48$, $72$, $96$, and $120$

We study degree-$2$ Siegel theta series of extremal even unimodular lattices from the genus-$2$ viewpoint initiated by Ozeki. Using Igusa's structure theorem, we define a depth filtration on genus-$2$ cusp forms, measured by the total degree in $χ_{10}$ and $χ_{12}$, and relate it to the vanishing of low Fourier--Jacobi coefficients forced by extremality. In ranks $48$, $72$, and $96$, this interaction closes exactly and yields a direct genus-$2$ proof that the degree-$2$ theta series is uniquely determined by extremality (conditional on existence in rank $96$). In rank $120$ (again conditional on existence), the same argument leaves a one-dimensional residual line spanned by $χ_{10}^6$: with $χ_{10}$ in the standard integral normalization, any two such degree-$2$ theta series differ by an integer multiple of $χ_{10}^6$.

math.NT↗
arXiv2026

Majorization and Gaussian-Mass Maximality for Construction-A Lattices from Binary Self-Dual Codes

Regev and Stephens-Davidowitz conjectured that the integer lattice maximizes Gaussian mass among integral lattices of a given rank. We prove this, including the equality case, for all unimodular Construction-A lattices arising from binary self-dual codes. The proof reduces the theta-series inequality to a sharp majorization statement for codes: if $C$ is a binary self-dual $[2k,k]$ code, then the half-weight distribution of $C$ is dominated in convex order by $\operatorname{Bin}(k,1/2)$, which is the corresponding distribution for the repetition-code model of $\mathbb{Z}^{2k}$. Indeed, after putting $C$ in systematic form $[I\mid A]$, self-duality gives $AA^T=I$ over $\mathbb{F}_2$, so for a uniformly random message $a$ the two weights $\operatorname{wt}(a)$ and $\operatorname{wt}(aA)$ have the same binomial law. The half-weight of the resulting codeword is their average, and Jensen's inequality then gives convex-order domination. Applied to the convex test functions that build the theta series, this yields a sum-of-squares formula for the Gaussian-mass gap; applied to hinge functions, it gives coefficientwise nonnegativity of the reduced gap polynomial.

math.NT↗
arXiv2026

Lines in the prime number graph

The prime number graph is the set of points $(n,p_n)$ where $p_n$ denotes the $n^{\rm th}$ prime. Let $L(n)$ be the minimum number of straight line segments needed to cover the first $n$ points in this set. Let $B(n)$ be the largest number of points $(k,p_k)$ with $k\le n$ covered by a single line. Recently Sloane conjectured that $L(n) = O(n/\log n)$. We show that $L(n)=O(n \log \log n / \log n)$ and $B(n)\ge c\log n$ for a constant $c>0$ and all large $n$. Under RH we show that for large $n$ we have $B(n)=O(n^{3/4}(\log n)^{1/2})$ and $ L(n)\ge c' n^{1/4} (\log n) ^{-1/2}$ for some constant $c'>0.$

math.NT↗
arXiv2026

A Sharp Reverse Minkowski Inequality for the Gaussian Mass of Integral Unimodular Lattices Through Rank $32$

The integer lattice $\mathbb{Z}^n$ is conjectured to maximize the Gaussian mass $Θ_L(t)=\sum_{x\in L}e^{-t\|x\|^2}$ over the set of stable lattices in $\mathbb{R}^n$, for every $t>0$. We prove this sharp inequality for every integral unimodular lattice $L$ of rank $n\leq 32$, with equality only at $L\cong\mathbb{Z}^n$, and furthermore obtain the strict inequality for every even unimodular lattice of rank $40$. The proof does not use the classification of unimodular lattices in these ranks; rather, it parametrizes integral unimodular theta series as polynomials in the modular function $u=Δ_8/\vartheta_3^8\in(0,1/64]$, with the few coefficients that arise controlled by norm-$1$ splitting, ADE root counts, and shadow positivity.

math.NT↗
arXiv2026

Saturation and No-Go Theorems for Scalar Poisson Certificates of Gaussian Mass Maximality

Regev and Stephens-Davidowitz conjectured that the Gaussian mass $Θ_Λ(t) = \sum_{x \in Λ} e^{-t\lVert x\rVert^2}$ of any integral lattice $Λ\subset \mathbb{R}^n$ is bounded above by $Θ_{\mathbb{Z}^n}(t)$. For $n\ge 4$, we prove a saturation theorem for the natural scalar Poisson-summation certificates of this conjecture: any such certificate that is sharp at $\mathbb{Z}^n$ must interpolate the Gaussian, and have vanishing Fourier transform, at every nonzero point of integer squared norm. Applied to the lattice $E_8 \oplus \mathbb{Z}^{n-8}$, this rigidity is incompatible with the strict theta-series gap $Θ_{\mathbb{Z}^8}(t) - Θ_{E_8}(t) = θ_2(it/π)^4\,θ_4(it/π)^4 > 0$. Consequently, in dimensions $n \ge 8$, no scalar Poisson certificate can attain the sharp $\mathbb{Z}^n$ Gaussian mass bound. The same argument rules out the corresponding scalar certificate strategy for the stable-lattice formulation of the conjecture, and extends to orbit-constant graded families $Λ\mapsto h_Λ$; near-sharp sequences are similarly excluded under a uniform summability hypothesis.

math.NT↗
arXiv2026

Equality in a Reverse Minkowski Shell Bound for Integral Lattices via Spherical Designs

For a full-rank integral lattice $\mathcal{L}\subset\mathbb{R}^n$, Regev and Stephens-Davidowitz proved that \[N_{=k}(\mathcal{L}):=|\{y\in\mathcal{L}:\lVert y\rVert^2=k\}|\le 2\binom{n+2k-2}{2k-1}.\] We classify the equality cases. For $n\ge2$, equality holds if and only if either $k=1$ and $\mathcal{L}\cong\mathbb{Z}^n$, or $n=8$, $k=2$, and $\mathcal{L}\cong E_8$. For $n=1$, equality holds exactly when $\mathcal{L}$ represents $k$. The proof shows that equality is rigid. Saturation of the shell bound forces the normalized norm-$k$ shell to be an antipodal tight spherical $(4k-1)$-design. The associated Delsarte--Goethals--Seidel annihilator polynomial gives an arithmetic root condition, which isolates $E_8$ at $k=2$, rules out $k=3$, and combines with the Bannai--Damerell/Bannai theorem and an elementary circle argument to exclude all remaining cases in dimension at least $2$.

math.NT↗
arXiv2026

Squared edge lengths of regular simplices with rational vertices

We determine exactly which positive rational numbers occur as squared edge lengths of regular $d$-simplices with vertices in $\mathbb{Q}^n$. The answer exhibits a sharp stabilization phenomenon: once $n-d\geq 3$, every positive rational number occurs, while codimensions $0$, $1$, and $2$ are governed by explicit square-class, norm-group, and Hilbert-symbol conditions. The proof reduces simplex realizability to the Hasse--Minkowski classification of rational quadratic forms.

math.NT↗
arXiv2026

How Thick Is the Sierpiński Triangle?

Although the Sierpiński triangle has planar area $0$, it is uniformly non-flat: at every point and every scale, its nearby points span a two-dimensional region of comparable size. We prove a sharp version of this statement, showing that the Feng--Wu thickness of $E$ is exactly $\sqrt{3}/6$, the inradius of a unit equilateral triangle. More precisely, if $E$ is the standard Sierpiński triangle of side length $1$ and $B(x,r)$ denotes the closed disk of radius $r$ centered at $x$, then for every $x\in E$ and every $0<r\le 1$, the convex hull of $E\cap B(x,r)$ contains an equilateral triangle of side length $r$. Consequently, $\operatorname{conv}(E\cap B(x,r))$ contains a closed disk of radius $(\sqrt{3}/6)r$; this constant is best possible. The proof is elementary -- boundary edges of all construction triangles survive in the limit set, and self-similarity reduces the problem to the normalized range $1/2\le r\le 1$.

math.MG↗
arXiv2026

A Fixed-Prime Criterion for Reciprocals in Missing-Digit Sets

We prove a structural upper bound on the $p$-adic valuation of denominators of rationals belonging to a missing-digit set $K_{m,D}$, generalizing a key step in recent work of Lin, Wu, and Yang [arXiv:2603.24614] on reciprocals of factorials. For a rational $\frac{r}{Q}$ with $\gcd(Q,m)=1$ and a fixed prime $p_0\nmid m$, membership in $K_{m,D}$ forces $ν_{p_0}(Q)$ to be controlled by the $p_0$-adic valuation of the multiplicative order of $m$ modulo the radical of $Q$, with explicit overhead depending only on $m$ and $D$. Because the obstruction is stated at the level of a single denominator, a pair of sequence-specific valuation estimates converts it into an effective finiteness criterion for $\left\{\frac{1}{a_n}:n\in\mathbb{N}\right\}\cap K_{m,D}$. Specializing to the case in which $Q$ is the part of $n!$ coprime to $m$ recovers the fixed-prime step in the Lin--Wu--Yang argument. As applications, we treat reciprocals of superfactorials, products of polynomial values, and products of Fibonacci numbers. We also exhibit an exponential family -- products of $(m^k-1)$ -- for which the full structural criterion applies but a coarser largest-prime-factor formulation does not.

math.NT↗
arXiv2026

An improved bound for sumsets of thick compact sets via the Shapley--Folkman theorem

Let $E_1,\dots,E_n \subset \mathbb{R}^d$ be compact sets of positive diameter with Feng--Wu thickness at least $c>0$. Feng and Wu proved that $E_1+\cdots+E_n$ has non-empty interior when $n>2^{11}c^{-3}+1$. We show that \[n>\frac{\sqrt d}{(\sqrt{1+c}-1)^2}=\frac{\sqrt d\,(\sqrt{1+c}+1)^2}{c^2}\] already suffices. In particular, since $0 6\sqrt d\,c^{-2}$ is enough. For fixed dimension $d$, this improves the exponent in $c^{-1}$ from $3$ to $2$, while introducing only an explicit factor of $\sqrt d$. The proof replaces the one-summand-at-a-time enlargement of Feng--Wu by a simultaneous convexification step based on a radius form of the Shapley--Folkman theorem.

math.MG↗