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Scott Duke Kominers

Publications and source records attributed to Scott Duke Kominers.

At least 19 recordsLinked to original sources
arXiv2026

On digit-appending walks along linear recurrences

We study digit-appending walks in base $b\geq 2$ along integer linear recurrences with minimal characteristic polynomials having a unique simple dominant root $α$ that lies outside the unit circle. We prove that if $α$ and $b$ are multiplicatively independent, then no infinite walk can have uniformly bounded appended digit lengths. In contrast, we show that when the appended digit length can be unbounded, the same independence hypothesis implies the existence of infinite walks.

math.NT↗
arXiv2026

A lattice family with kissing numbers $τ(\mathcal{L}_n) \ge e^{2 \sqrt{n}}$

For all prime powers $q\geq5$, we construct lattices $\mathcal{L}_q\subseteq\mathbb{Z}^q$ with kissing numbers \[ τ(\mathcal{L}_q)\geq \left(\frac{1}{2πe^2}+o(1)\right)\sqrt{q}\,e^{2\sqrt{q}}. \] The same asymptotic bound holds on a set of integer dimensions of natural density $1$, and in every sufficiently large integer dimension $n$ with an additional factor $e^{-\tfrac{1}{2}n^{1/40}}$. The construction is an extension of a previous construction by Bennett-Peikert based on Reed-Solomon codes.

math.CO↗
arXiv2026

Breaking the Fourth Wall: The Planar Centroid Banach--Mazur Diameter Is Strictly Less than $4$

We prove that the centroid Banach--Mazur diameter of planar convex bodies is strictly less than $4$, (slightly) improving the prior upper bound $69/17\approx 4.05882$ due to Lassak. The proof combines a sharp covariance-ellipse sandwich, whose equality cases in any single direction occur only for triangles, with a compactness argument: an extremal pair at distance $4$ would consist of two triangles, which are linearly equivalent and hence must in fact be at distance $1$. The resulting gap below $4$ is uniform but nonquantitative.

math.MG↗
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↗