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Boris Alexeev

Publications and source records attributed to Boris Alexeev.

At least 19 recordsLinked to original sources
arXiv2026

Primitive sets and von Mangoldt chains: Erd\H{o}s Problem #1196 and beyond

A set of integers is primitive if no number in the set divides another. We introduce a new method for bounding Erd\H{o}s sums of primitive sets, suggested from output of GPT-5.4 Pro, based on Markov chains with von Mangoldt weights. The method leads to a host of applications, yet seems to have been overlooked by the prior literature since Erd\H{o}s's seminal 1935 paper. As applications, we prove two 1966 conjectures of Erd\H{o}s-S\'ark\"ozy-Szemer\'edi, on primitive sets of large numbers (#1196) and on divisibility chains (#1217). The method also provides a short proof of the Erd\H{o}s Primitive Set Conjecture (#164), as well as the related claim that 2 is an ''Erd\H{o}s-strong'' prime. Moreover, the method resolves a revised form of the Banks-Martin conjecture, which has long been viewed as a unifying `master theorem' for the area.

math.NT↗
arXiv2026

Short proofs in combinatorics, probability and number theory II

We give a quintet of proofs resulting from questions posed by Erd\H{o}s. These questions concern ordinary lines in planar point sets, sequences with uniformly small exponential sums, $K_4$-free $4$-critical graphs with few chords in any cycle, a counterexample to a "fewnomial" version of the Erd\H{o}s--Tur\'{a}n discrepancy bound, and a finiteness theorem for integers $n$ such that $n-a k^2$ is prime for all $k\leq \sqrt{n/a}$ coprime to $n$ (for fixed $a\in\mathbb Z_+$). Each proof is due to an internal model at OpenAI.

math.CO↗
arXiv2026

Short proofs in combinatorics and number theory

We give a triplet of short proofs, each of which answers a question raised by Erd\H{o}s. The first concerns the small prime factors of $\binom{n}{k}$, the second concerns whether an additive basis $A$ can always be split into pieces $A_1$ and $A_2$ such that each of $A_i + A_i$ has bounded gaps, and the final concerns whether $\{\alpha p\}$ is "well-distributed" in the sense introduced by Hlawka and Petersen. In each case, the proof is due entirely to an internal model at OpenAI.

math.CO↗
arXiv2025

Asymptotically optimal approximate Hadamard matrices

An approximate Hadamard matrix is a well-conditioned square matrix with all entries in $\{\pm1\}$. We measure the quality of a matrix by its condition number, i.e., the ratio of its largest and smallest singular values. We prove that for any fixed positive $\alpha<17/92$, every sufficiently large dimension admits an approximate Hadamard matrix with condition number at most $1+n^{-\alpha}$. In particular, the smallest possible condition number tends to $1$ as $n\to\infty$. Conversely, there exists an absolute constant $c>0$ such that for every sufficiently large $n\not\equiv0\pmod4$, every $n\times n$ matrix with entries in $\{\pm1\}$ has condition number at least $1+c(\log n)/n$. Along the way, we resolve a problem of Jaming and Matolcsi concerning flat orthogonal matrices, and we conclude by describing several explicit infinite families of approximate Hadamard matrices.

math.CO↗
arXiv2025

Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof

We resolve a $1000 Erd\H{o}s prize problem, complete with formal verification generated by a large language model. In over a dozen papers, beginning in 1976 and spanning two decades, Paul Erd\H{o}s repeatedly posed one of his "favourite" conjectures: every finite Sidon set can be extended to a finite perfect difference set. We establish that {1, 2, 4, 8, 13} is a counterexample to this conjecture. During the preparation of this paper, we discovered that although this problem was presumed to be open for half a century, Marshall Hall, Jr. published a different counterexample three decades before Erd\H{o}s first posed the problem. With a healthy skepticism of this apparent oversight, and out of an abundance of caution, we used ChatGPT to vibe code a Lean proof of both Hall's and our counterexamples.

math.CO↗
arXiv2025

The independence and clique cover numbers of the squarefree graph

We determine the largest subset $A\subseteq \{1,\dotsc,n\}$ such that for all $a,b\in A$, the product $ab$ is not squarefree. Specifically, the maximum size is achieved by the complement of the odd squarefree numbers. This resolves a problem of Paul Erd\H{o}s and Andr\'as S\'ark\"ozy from 1992.

math.CO↗
arXiv2025

Chunking Attacks on File Backup Services using Content-Defined Chunking

Systems such as file backup services often use content-defined chunking (CDC) algorithms, especially those based on rolling hash techniques, to split files into chunks in a way that allows for data deduplication. These chunking algorithms often depend on per-user parameters in an attempt to avoid leaking information about the data being stored. We present attacks to extract these chunking parameters and discuss protocol-agnostic attacks and loss of security once the parameters are breached (including when these parameters are not setup at all, which is often available as an option). Our parameter-extraction attacks themselves are protocol-specific but their ideas are generalizable to many potential CDC schemes.

cs.CR↗
arXiv2025

Decomposing a factorial into large factors

Let $t(N)$ denote the largest number such that $N!$ can be expressed as the product of $N$ integers greater than or equal to $t(N)$. The bound $t(N)/N = 1/e-o(1)$ was apparently established in unpublished work of Erd\H{o}s, Selfridge, and Straus; but the proof is lost. Here we obtain the more precise asymptotic $$ \frac{t(N)}{N} = \frac{1}{e} - \frac{c_0}{\log N} + O\left( \frac{1}{\log^{1+c} N} \right)$$ for an explicit constant $c_0 = 0.30441901\dots$ and some absolute constant $c>0$, answering a question of Erd\H{o}s and Graham. For the upper bound, a further lower order term in the asymptotic expansion is also obtained. With numerical assistance, we obtain highly precise computations of $t(N)$ for wide ranges of $N$, establishing several explicit conjectures of Guy and Selfridge on this sequence. For instance, we show that $t(N) \geq N/3$ for $N \geq 43632$, with the threshold shown to be best possible.

math.NT↗
arXiv2025

On the clustering behavior of sliding windows

Things can go spectacularly wrong when clustering timeseries data that has been preprocessed with a sliding window. We highlight three surprising failures that emerge depending on how the window size compares with the timeseries length. In addition to computational examples, we present theoretical explanations for each of these failure modes.

cs.LG↗
arXiv2025

Rewinding the byte trail of the White Whale

Motivated by a popular code golf challenge, we review some key ideas from information theory and discuss how to efficiently compress a streaming file with an acceptable error rate.

cs.IT↗
arXiv2024

The Erd\H{o}s unit distance problem for small point sets

We improve the best known upper bound on the number of edges in a unit-distance graph on $n$ vertices for each $n\in\{16,\ldots,30\}$. When $n\leq 21$, our bounds match the best known lower bounds, and we fully enumerate the densest unit-distance graphs in these cases. On the combinatorial side, our principle technique is to more efficiently generate $\mathcal{F}$-free graphs for a set of forbidden subgraphs $\mathcal{F}$. On the algebraic side, we are able to determine programmatically whether many graphs are unit-distance, using a custom embedder that is more efficient in practice than tools such as cylindrical algebraic decomposition.

math.CO↗
arXiv2024

The nerd snipers problem

We correct errors that appear throughout "The vicious neighbour problem" by Tao and Wu. We seek to solve the following problem. Suppose $N$ nerds are distributed uniformly at random in a square region. At 3:14pm, every nerd simultaneously snipes their nearest neighbor. What is the expected proportion $P_N$ of nerds who are left unscathed in the limit as $N\to\infty$?

math.MG↗
arXiv2024

The Penults of Tak: Adventures in impartial, normal-play, positional games

For normal play, impartial games, we define penults as those positions in which every option results in an immediate win for the other player. We explore the number of tokens in penults of two positional games, Impartial Tic and Impartial Tak. We obtain a complete classification in the former case. We then explore winning strategies and further directions.

math.CO↗
arXiv2020

The Base Measure Problem and its Solution

Probabilistic programming systems generally compute with probability density functions, leaving the base measure of each such function implicit. This mostly works, but creates problems when densities with respect to different base measures are accidentally combined or compared. Mistakes also happen when computing volume corrections for continuous changes of variables, which in general depend on the support measure. We motivate and clarify the problem in the context of a composable library of probability distributions and bijective transformations. We solve the problem by standardizing on Hausdorff measure as a base, and deriving formulas for comparing and combining mixed-dimension densities, as well as updating densities with respect to Hausdorff measure under diffeomorphic transformations. We also propose a software architecture that implements these formulas efficiently in the common case. We hope that by adopting our solution, probabilistic programming systems can become more robust and general, and make a broader class of models accessible to practitioners.

cs.PL↗
arXiv2017

Partisan gerrymandering with geographically compact districts

Bizarrely shaped voting districts are frequently lambasted as likely instances of gerrymandering. In order to systematically identify such instances, researchers have devised several tests for so-called geographic compactness (i.e., shape niceness). We demonstrate that under certain conditions, a party can gerrymander a competitive state into geographically compact districts to win an average of over 70% of the districts. Our results suggest that geometric features alone may fail to adequately combat partisan gerrymandering.

math.CO↗
arXiv2017

An impossibility theorem for gerrymandering

The U.S. Supreme Court is currently deliberating over whether a proposed mathematical formula should be used to detect unconstitutional partisan gerrymandering. We show that in some cases, this formula will only flag bizarrely shaped districts as potentially constitutional.

math.CO↗
arXiv2013

Phase retrieval with polarization

In many areas of imaging science, it is difficult to measure the phase of linear measurements. As such, one often wishes to reconstruct a signal from intensity measurements, that is, perform phase retrieval. In this paper, we provide a novel measurement design which is inspired by interferometry and exploits certain properties of expander graphs. We also give an efficient phase retrieval procedure, and use recent results in spectral graph theory to produce a stable performance guarantee which rivals the guarantee for PhaseLift in [Candes et al. 2011]. We use numerical simulations to illustrate the performance of our phase retrieval procedure, and we compare reconstruction error and runtime with a common alternating-projections-type procedure.

cs.IT↗