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Samuel Korsky

Publications and source records attributed to Samuel Korsky.

At least 19 recordsLinked to original sources
arXiv2026

A Resolution of the de Bruijn--Erd\H{o}s Consecutive-Gap Problem

Let $(x_n)_{n\geq1}$ be a sequence of distinct points on the unit circle. An $r$-span is the total length of $r$ consecutive gaps determined by the inserted points. Write $M_n^{(r)}$ and $m_n^{(r)}$ for the largest and smallest $r$-spans after the first $n$ insertions. We prove that there is an absolute constant $c>0$ such that, for every sufficiently large $r$, \[ \limsup_{n\to\infty}\bigl(nM_n^{(r)}-r\bigr) \geq c\sqrt{\log r}, \qquad \limsup_{n\to\infty}\bigl(r-nm_n^{(r)}\bigr) \geq c\sqrt{\log r}, \] and \[ \limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}} \geq 1+\frac{\log r}{100r}. \] Thus all three asymptotic conjectures made by de Bruijn and Erd\H{o}s in 1949 are resolved. The ratio bound matches the upper bound of Cl\'ement and Steinerberger up to an absolute constant and answers a question of Brethouwer. The proofs compare interval counts at nearby times. Pointwise control leads to a one-dimensional sequence-discrepancy argument for the ratio, while averaged control and Hal\'asz's planar $L^1$ discrepancy theorem give the two one-sided conclusions.

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arXiv2026

Affine Copies of Three-Point Patterns in Sets of Integers

Let $P=\{0,a,b\}$, where $0 0$, and let $M_P(A)$ count the copies with any $d\ne0$. We prove that every such three-point pattern other than the arithmetic progression $\{0,1,2\}$ satisfies \[ M_P^+(A)\le \frac{99}{400}|A|^2+O(|A|), \qquad M_P(A)\le \frac{13}{28}|A|^2+O_P(|A|). \] For the particular pattern $P=\{0,1,3\}$ -- the subject of a question raised by Ganguly and recorded as Problem 24 in Green's list of open problems -- we sharpen the bound allowing both signs of the dilation to \[ M_{\{0,1,3\}}(A)\le \frac{47}{122}|A|^2+O(|A|). \]

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arXiv2026

Long Lattice Paths with No Three Collinear Vertices

For $d\ge 1$, let $L(d)\in\mathbb N\cup\{\infty\}$ be the supremum of the lengths of paths in $\mathbb Z^d$ whose steps are standard basis vectors and whose vertex sets contain no collinear triple. We prove that \[ \log_2\log_2 L(d)\ge \frac{2}{5}d-O(1) \] for all sufficiently large $d$.

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arXiv2026

Positive Lower Density for Hofstadter's $ab-1$ Problem

Let $A$ be the smallest set of positive integers containing $2$ and $3$ such that $ab-1\in A$ whenever $a,b\in A$ are distinct. We prove that $A$ has positive lower density, answering a problem of Erd\H{o}s attributed to Hofstadter.

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arXiv2026

Asymptotically attaining the Moore bound

For positive integers $d$ and $k$, let $n_k(d)$ be the maximum order of a graph of maximum degree at most $d$ and diameter at most $k$. We prove that $$ \lim_{d\to\infty}\frac{n_k(d)}{d^k}=1$$ for every fixed $k$, thereby resolving the asymptotic degree-diameter problem for fixed diameter and proving a conjecture of Bollob\'as. The lower bound comes from regular graphs $H_{k,q}$, indexed by prime powers $q$, whose vertices are partial flags in $\mathbb{F}_q^{\,2k+1}$. These graphs have diameter $k$ and order $|V(H_{k,q})| =(1+o(1))\Delta(H_{k,q})^k$. We also construct, for every fixed $\ell \ge 2$, graphs of maximum degree at most $d$ and line-graph diameter at most $\ell$ with $(1+o(1))d^{\ell}$ edges.

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arXiv2026

Improved Bounds for Distinct Multiples in Intervals

For $n\ge1$, let $F(n)$ be the least $H$ such that any $H$ consecutive integers contain $n$ pairwise distinct integers $a_1, a_2, \dots, a_n$ with $k \mid a_k$ for $1\le k\le n$, and define $h_{\mathbb P}(n)$ analogously for the primes at most $n$. We prove \[ F(n)\le n^{4/3}\exp\!\left(O\!\left(\frac{\log n}{\log\log n}\right)\right), \qquad h_{\mathbb P}(n)\ll \frac{n^{4/3}}{(\log n)^{1/3}}, \] and \[ F(n) \ge h_{\mathbb P}(n)\ge n\exp\!\left( \left(\frac{\log 2}{2}-o(1)\right) \frac{\log n}{\log\log n} \right). \] The upper bounds follow from a new estimate for unions of arithmetic progressions. The lower bound adapts a quadratic-residue compression construction of Green and Ruzsa.

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arXiv2026

Large Sets of Integers with No Harmonic Triples

Let $f(N)$ denote the largest size of a set $A\subseteq [N]=\{1,\ldots,N\}$ containing no distinct $a,b,c$ such that \[ \frac2a=\frac1b+\frac1c . \] We prove \[ f(N)\gg N\exp\!\left(-(2\sqrt{\log(24/7)}+o(1))\sqrt{\log\log N}\right). \] The construction filters the odd integers up to $N$ by a random affine image of a dense three-term-progression-free set in a prime field $\mathbb{F}_q$ with $q\asymp\log N$, and then deletes a controlled family of collapsed triples.

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arXiv2026

A Stretched-Exponential Bound for an Erdos--Graham Unit-Fraction Problem

For a finite multiset $A$ of positive integers, write $\mathcal{R}(A)=\sum_{a\in A}a^{-1}$ and let $\varepsilon(A)$ be the distance from $1$ to the largest reciprocal subsum of $A$ that does not exceed $1$. Erd\H{o}s and Graham proved that $\varepsilon(A)\ll K^{-2}$ whenever $\mathcal{R}(A)>K$, and asked whether one always has $\varepsilon(A)\leq \exp(-cK)$ for an absolute constant $c>0$. We prove the stretched-exponential estimate $$ \varepsilon(A)\leq \exp\bigl(-c\sqrt{K\log K}\bigr) $$ for all sufficiently large $K$.

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arXiv2026

North-East Lattice Paths with Few Collinear Vertices

Let $A(k)$ be the largest possible number of moves in a north-east lattice path whose visited vertices contain no $k$ collinear points. Gerver (1979) and Gerver and Ramsey (1979) gave lower and upper bounds on $A(k)$ of the form \[ \exp\left(\Omega(\log(k)^2)\right)\le A(k)\le \exp\left(O(k^4)\right). \] Improving upon these results, we show that \[ \exp\left(\Omega(k^{1/3})\right)\le A(k)\le \exp\left(\left(\frac{2}{e}+o(1)\right)(k-1)^2\right). \]

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arXiv2026

Arithmetic Progression-Free Subset-Sum Sets

For a finite set $A$ of positive integers, let $H(A)$ be its set of subset sums, including the empty sum, and let $g_k(n)$ be the least $N$ for which some $n$-element set $A\subseteq[N]$ has $H(A)$ free of nonconstant $k$-term arithmetic progressions. The problem of determining $g_k(n)$ was posed by Erd\H{o}s and S\'ark\H{o}zy. In the three-term case, we prove a lower bound equal to the exact bandwidth of the ternary grid. If $T_m=[x^m](1+x+x^2)^m$ is the central trinomial coefficient, then \[ g_3(n)\ge \frac{T_n-1}{2}+\sum_{j=0}^{n-1}T_j =\left(\frac{\sqrt{3}}{2\sqrt{\pi}}+o(1)\right)\frac{3^n}{\sqrt{n}}. \] For general $k \ge 4$ we show \[ g_k(n)\gg_k \left(\frac{k-1}{k-2}\right)^n n^{-\log_2((k-1)/(k-2))} \] In the opposite direction, a carry-free digit construction based on nearly-regular graphs gives \[ \limsup_{n\to\infty}g_k(n)^{1/n} \le \min_{p\ \mathrm{prime},\ p\ge3}p^{2/(\min\{p,k\}-1)}. \] Consequently, as $k\to\infty$, the logarithm of the lower exponential rate is at least $(1+o(1))/k$, while the logarithm of the upper exponential rate is at most $(2+o(1))\log k/k$.

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arXiv2026

Hitting Arithmetic Progressions at the Square-Root Scale

For positive integers $N$ and $k$, let $f(N,k)$ be the minimum size of a set $A\subseteq\{0,1,\ldots,N-1\}$ which intersects every $k$-term arithmetic progression contained in $\{0,1,\ldots,N-1\}$. Brown and Freedman introduced this hitting problem for arithmetic progressions and studied it for growing $k$. The square-root scale $k=\sqrt N$ is a natural transition point. Truss proved \[ f(n^2,n)>n+\frac12 n^{1/2}-2. \] We improve the leading constant in the second-order term, proving \[ f(n^2,n)\ge n+\left(\frac1{\sqrt2}+o(1)\right)n^{1/2}. \] On the upper-bound side, Brown and Freedman proved $f(p^2,p)\le 2p-2$ for odd primes $p$, and subsequent Szekeres-type constructions give logarithmic savings. We prove the stronger asymptotic upper bound \[ f(p^2,p) \le 2p-\left(\sqrt{\frac23}-o(1)\right)\sqrt{\frac p{\log p}} \] for sufficiently large prime $p$. The upper bound is obtained by a randomized front construction with an alteration step.

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arXiv2026

An Improved Lower Bound for the de Bruijn--Erd\H{o}s Consecutive Gap Problem

Let $(x_n)_{n\geq 1}$ be a sequence of distinct points on the unit circle. After the first $n$ points are inserted, the circle is divided into $n$ intervals. For a fixed integer $r\geq 1$, let $M_n^{(r)}$ and $m_n^{(r)}$ denote respectively the largest and smallest total lengths of $r$ consecutive intervals. A theorem of de Bruijn and Erd\H{o}s gives \[ \limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}}\geq 1+\frac1r . \] The case $r=1$ is sharp and gives the classical factor $2$. The cases $r\geq 2$ remain much less understood. We prove the improved lower bound \[ \limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}} \geq 1+\frac{r}{r^2-1} \qquad (r\geq 2). \] In particular, for two consecutive intervals the lower bound becomes $5/3$, improving the de Bruijn--Erd\H{o}s bound $3/2$.

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arXiv2026

Inhomogeneous Approximation by Sums of Roots

Let $d\geq 2$ and $k\geq 1$ be fixed. We prove that, for every $\epsilon>0$ and every real $\beta$, there exist integers $1\leq b_1,\ldots,b_k\leq N$ such that \[ \left\|\sum_{j=1}^k b_j^{1/d}-\beta\right\| \ll_{d,k,\epsilon} N^{-k/d+\epsilon}. \] The proof combines Schmidt's Subspace Theorem with an explicit inhomogeneous transference argument. This improves Iyer's (2025) higher-root exponent $(k-d+1)/d^2$, and also the analogous $d$-ary full-basis exponent away from the cases where $k+1$ is a power of $d$, at the cost of ineffectivity. We also record a conjectural uniform exponent $k-1/d$. In the square-root case $d=2$, we give explicit integer-target constructions for $k=2,3,4$ attaining this conjectural value.

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arXiv2026

Lipschitz Functions on Sparse Graphs II

Korsky, Saffat and Aiylam introduced a growth constant $c(G)$ for integer-valued $h$-Lipschitz functions on a finite graph $G$ and proved that, for $G=G(n,d/n)$, \[ \frac{1}{2d}+O(d^{-2})\le \log c(G)\le \frac{4\log^2 d}{d}+O(d^{-1}) \] with high probability. We sharpen the random-graph part of their result; as $n\to\infty$ and then $d\to\infty$, we prove \[ \log c(G)=\frac{\pi^2}{6d}+o(d^{-1}) \] with high probability. Additionally, we derive bounds on $\log c(Q_d)$ where $Q_d$ is the $d$-dimensional hypercube graph: \[ \frac{\pi^2}{6d}+o(d^{-1}) \le \log{c(Q_d)}\le \left(\frac{3}{4} + o(1)\right)\frac{\log d}{d}. \]

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arXiv2026

Polylogarithmic Full-Chord Buffon Discrepancy

Steinerberger introduced the Buffon discrepancy problem, asking how accurately a one-dimensional set of length $L$ in a convex body can match the Crofton-predicted line-intersection counts, and proved an $O\left(L^{1/3}\right)$ upper bound via a Steinhaus longimeter construction. Using the Aistleitner--Bilyk--Nikolov arbitrary-measure star-discrepancy theorem we demonstrate the existence of full-chord constructions with discrepancy $O\left((\log L)^{3/2}\right)$ for every fixed compact convex body with finite piecewise $C^2$ boundary. In the disk, we prove that every full-chord construction has discrepancy at least $\Omega\left(\log L\right)$, using Schmidt's two-dimensional rectangle discrepancy lower bound.

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arXiv2026

Randomly Shifted Steinhaus Longimeters and Buffon Discrepancy

Let $\Omega \subset \mathbb{R}^2$ be a bounded convex domain. Steinerberger (2026) introduced the Buffon discrepancy problem: given length $L$, construct a one-dimensional set $S\subset\Omega$ such that the number of intersections of $S$ with a line $\ell$ approximates the Crofton-normalized chord length $$ \frac{2L}{\pi|\Omega|}\cdot\mathcal{H}^1(\ell\cap\Omega).$$ Steinerberger proved a universal upper bound of order $L^{1/3}$ using a Steinhaus longimeter construction, and showed that the disk admits bounded discrepancy. We prove that a randomly shifted Steinhaus construction improves the order of the universal upper bound to $L^{1/5}(\log L)^{2/5}$.

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arXiv2026

Spectral Bounds for Antipodal Graphs

Suppose $\left\{x_1, \dots, x_n\right\} \subset \mathbb{R}^2$ is a set of $n$ points in the plane with diameter $\leq 1$, meaning $|x_i - x_j| \leq 1$ for all $1 \leq i,j \leq n$. We show that the ratio of the number of ``neighbors'' (ordered pairs of points with distance $\leq \varepsilon$) to the number of ``antipodes'' (ordered pairs of points with distance $\geq 1 - \varepsilon$) is $\gtrsim\varepsilon^{1/2 + o(1)}$, attaining the conjectured correct asymptotic within a polylog factor and improving the $\gtrsim\varepsilon^{3/4+o(1)}$ bound of Steinerberger (2025). In dimensions $d\ge3$ we prove a similar result with exponent $3(d - 1)/4$.

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arXiv2024

Permutations of counters on a table

We consider a game in which a blindfolded player attempts to set $n$ counters lying on the vertices of a rotating regular $n$-gon table simultaneously to $0$. When the counters count$\pmod{m}$ we simplify the argument of Bar Yehuda, Etzion, and Moran (1993) showing that the player can win if and only if $n = 1$, $m = 1$, or $(n, m) = (p^a, p^b)$ for some prime $p$ and $a, b \in \mathbb{N}$. We broadly generalize the result to the setting where the counters can be permuted by any element of a subset of the symmetric group $S \subseteq S_n$, with the original formulation corresponding to $S = \mathbb{Z}_n$ (rotations of the table).

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