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Nat Sothanaphan

Publications and source records attributed to Nat Sothanaphan.

14 recordsLinked to original sources
arXiv2026

Resolution of Erd\H{o}s Problem #728: a writeup of Aristotle's Lean proof

We provide a writeup of a resolution of Erd\H{o}s Problem #728; this is the first Erd\H{o}s problem (a problem proposed by Paul Erd\H{o}s which has been collected in the Erd\H{o}s Problems website) regarded as fully resolved autonomously by an AI system. The system in question is a combination of GPT-5.2 Pro by OpenAI and Aristotle by Harmonic, operated by Kevin Barreto. The final result of the system is a formal proof written in Lean, which we translate to informal mathematics in the present writeup for wider accessibility. The proved result is as follows. We show a logarithmic-gap phenomenon regarding factorial divisibility: For any constants $0<C_1<C_2$ and $0 < \varepsilon < 1/2$ there exist infinitely many triples $(a,b,n)\in\mathbb N^3$ with $\varepsilon n \le a,b \le (1-\varepsilon)n$ such that \[ a!\,b!\mid n!\,(a+b-n)!\qquad\text{and}\qquad C_1\log n < a+b-n < C_2\log n. \] The argument reduces this to a binomial divisibility $\binom{m+k}{k}\mid\binom{2m}{m}$ and studies it prime-by-prime. By Kummer's theorem, $\nu_p\binom{2m}{m}$ translates into a carry count for doubling $m$ in base $p$. We then employ a counting argument to find, in each scale $[M,2M]$, an integer $m$ whose base-$p$ expansions simultaneously force many carries when doubling $m$, for every prime $p\le 2k$, while avoiding the rare event that one of $m+1,\dots,m+k$ is divisible by an unusually high power of $p$. These "carry-rich but spike-free" choices of $m$ force the needed $p$-adic inequalities and the divisibility. The overall strategy is similar to results regarding divisors of $\binom{2n}{n}$ studied earlier by Erd\H{o}s and by Pomerance.

math.NT↗
arXiv2025

An improved lower bound to Erdos' problem concerning products of distances for fixed diameter

Erdos, Herzog and Piranian asked whether, for $n$ points in the plane with fixed diameter (maximum distance between points), an arrangement of a regular $n$-gon maximizes their product of all pairs of distances. Recently, it was discovered that, for every even $n \geq 4$, a regular $n$-gon is not a maximizer. However, the discovered improvement turns out to be very small. Indeed, for a fixed diameter of $2$, let $\Delta$ be the square of the product of all pairs of distances (the "square" is here due to connections with polynomial discriminants). Then, for a regular $n$-gon, $\Delta = n^n$ for even $n$. The discovered arrangements have proven $\Delta = (1+o(1))n^n$ thus far, and it was not known whether one can have $\Delta \geq C n^n$ for some $C > 1$ and all sufficiently large even $n$. In this note, we show that indeed $\liminf_{n\to\infty} \Delta_{\max}/n^n > 1.037$ for even $n$ which settles this conjecture. Other arrangements with higher conjectured $\Delta/n^n$ values are in fact known, but we have not been able to obtain proofs that they have large products of distances. Finally, no arrangements such that $\Delta/n^n \to \infty$ are known and we do not know whether they exist.

math.MG↗
arXiv2025

Irregular Stanley sequences plausibly do not have growth $\Theta(n^2/\log n)$

Stanley sequences starting from the set $\{0, n\}$ where $n$ is a positive integer have long been conjectured to be divided into two types: the "regular" type where the growth rate is $\Theta(n^{\log_2(3)})$, and the "irregular" type where the growth rate is thought to be $\Theta(n^2/\log n)$. A paradigmatic case of a candidate irregular type is $n=4$, although to date no value of $n$ has been proven to have such a growth rate. Here, we provide strong numerical evidence against this conjectured growth rate for $n=4$. Specifically, for $n=4$, it seems plausible that the upper bound is $O(n^2/\log n)$ but that the lower bound is in fact $\Omega(n^{2-\delta})$ for some $\delta > 0$. This appears to be because the sequence is not totally "random" as has been assumed. Limitations of the numerical method here is discussed.

math.NT↗
arXiv2021

Riesz bases of exponentials and multi-tiling in finite abelian groups

Motivated by the open problem of exhibiting a subset of Euclidean space which has no exponential Riesz basis, we focus on exponential Riesz bases in finite abelian groups. We point out that that every subset of a finite abelian group has such a basis, removing interest in the existence question in this context. We then define tightness quantities for subsets to measure the conditioning of Riesz bases; for normalized tightness quantities, a value of one corresponds to an orthogonal basis, and a value of infinity corresponds to nonexistence of a basis. As an application, we obtain new weak evidence in favor of the open problem by giving a sequence of subsets of finite abelian groups whose tightness quantities go to infinity in the limit. We also prove that the Cartesian product of a set with a finite abelian group has the same tightness quantities as the original set. Lastly, under an additional hypothesis, explicit bounds are given for tightness quantities in terms of a subset's lowest multi-tiling level by a subgroup and its geometric configuration. This establishes a quantitative link between discrete geometry and harmonic analysis in this setting.

math.CO↗
arXiv2020

1D Triple Bubble Problem with Log-Convex Density

We prove that for a symmetric, strictly log-convex density on the real line, there are four possible types of perimeter-minimizing triple bubbles. This extends the work of Bongiovanni et al., which shows that there are two possible types of perimeter-minimizing double bubbles.

math.MG↗
arXiv2019

Fuglede's conjecture fails in 4 dimensions over odd prime fields

Fuglede's conjecture in $\mathbb{Z}_{p}^{d}$, $p$ a prime, says that a subset $E$ tiles $\mathbb{Z}_{p}^{d}$ by translation if and only if $E$ is spectral, meaning any complex-valued function $f$ on $E$ can be written as a linear combination of characters orthogonal with respect to $E$. We disprove Fuglede's conjecture in $\mathbb{Z}_{p}^{4}$ for all odd primes $p$, by using log-Hadamard matrices to exhibit spectral sets of size $2p$ which do not tile, extending the result of Aten et al. that the conjecture fails in $\mathbb{Z}_{p}^{4}$ for primes $p \equiv 3 \pmod 4$ and in $\mathbb{Z}_{p}^{5}$ for all odd primes $p$. We show, however, that our method does not extend to $\mathbb{Z}_{p}^{3}$. We also prove the conjecture in $\mathbb{Z}_{2}^{4}$, resolving all cases of four-dimensional vector spaces over prime fields. Our simple proof method does not extend to higher dimensions. The authors, however, have written a computer program to verify that the conjecture holds in $\mathbb{Z}_{2}^{5}$ and $\mathbb{Z}_{2}^{6}$. Finally, we modify Terry Tao's counterexample to show that the conjecture fails in $\mathbb{Z}_{2}^{10}$. Fuglede's conjecture in $\mathbb{Z}_{p}^{d}$ is now resolved in all cases except when $d=3$ and $p\geq 11$, or when $p=2$ and $d=7,8,9$.

math.NT↗
arXiv2019

The Least-Area Tetrahedral Tile of Space

We determine the least-area unit-volume tetrahedral tile of Euclidean space, without the constraint of Gallagher et al. that the tiling uses only orientation-preserving images of the tile. The winner remains Sommerville's type 4v.

math.MG↗
arXiv2018

Double Bubbles on the Real Line with Log-Convex Density

The classic double bubble theorem says that the least-perimeter way to enclose and separate two prescribed volumes in $\mathbb{R}^N$ is the standard double bubble. We seek the optimal double bubble in $\mathbb{R}^N$ with density, which we assume to be strictly log-convex. For $N=1$ we show that the solution is sometimes two contiguous intervals and sometimes three contiguous intervals. In higher dimensions, we think that the solution is sometimes a standard double bubble and sometimes concentric spheres (e.g. for one volume small and the other large).

math.MG↗
arXiv2018

Double Bubbles on the Line with Log-convex Density $f$ with $(\log f)'$ Bounded

We extend results of Bongiovanni et al. on double bubbles on the line with log-convex density to the case where the derivative of the log of the density is bounded. We show that the tie function between the double interval and the triple interval still exists but may blow up to infinity in finite time. For the first time, a density is presented for which the blowup time is positive and finite.

math.MG↗
arXiv2018

Naive Bayesian Learning in Social Networks

The DeGroot model of naive social learning assumes that agents only communicate scalar opinions. In practice, agents communicate not only their opinions, but their confidence in such opinions. We propose a model that captures this aspect of communication by incorporating signal informativeness into the naive social learning scenario. Our proposed model captures aspects of both Bayesian and naive learning. Agents in our model combine their neighbors' beliefs using Bayes' rule, but the agents naively assume that their neighbors' beliefs are independent. Depending on the initial beliefs, agents in our model may not reach a consensus, but we show that the agents will reach a consensus under mild continuity and boundedness assumptions on initial beliefs. This eventual consensus can be explicitly computed in terms of each agent's centrality and signal informativeness, allowing joint effects to be precisely understood. We apply our theory to adoption of new technology. In contrast to Banerjee et al. [2018], we show that information about a new technology can be seeded initially in a tightly clustered group without information loss, but only if agents can expressively communicate their beliefs.

cs.SI↗
arXiv2018

Isoperimetry in Surfaces of Revolution with Density

The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted volume with minimum weighted perimeter. According to Chambers' recent proof of the log-convex density conjecture, for many densities on $\mathbb{R}^n$ the answer is a sphere about the origin. We seek to generalize his results to some other spaces of revolution or to two different densities for volume and perimeter. We provide general results on existence and boundedness and a new approach to proving circles about the origin isoperimetric.

math.MG↗
arXiv2018

Determinants of Block Matrices with Noncommuting Blocks

Let $M$ be an $mn\times mn$ matrix over a commutative ring $R$. Divide $M$ into $m \times m$ blocks. Assume that the blocks commute pairwise. Consider the following two procedures: (1) Evaluate the $n \times n$ determinant formula at these blocks to obtain an $m \times m$ matrix, and take the determinant again to obtain an element of $R$; (2) Take the $mn \times mn$ determinant of $M$. It is known that the two procedures give the same element of $R$. We prove that if only certain pairs of blocks of $M$ commute, then the two procedures still give the same element of $R$, for a suitable definition of noncommutative determinants. We also derive from our result further collections of commutativity conditions that imply this equality of determinants, and we prove that our original condition is optimal under a particular constraint.

math.RA↗
arXiv2015

A Curved Brunn-Minkowski Inequality for the Symmetric Group

In this paper, we construct an injection $A \times B \rightarrow M \times M$ from the product of any two nonempty subsets of the symmetric group into the square of their midpoint set, where the metric is that corresponding to the conjugacy class of transpositions. If $A$ and $B$ are disjoint, our construction allows to inject two copies of $A \times B$ into $M \times M$. These injections imply a positively curved Brunn-Minkowski inequality for the symmetric group analogous to that obtained by Ollivier and Villani for the hypercube. However, while Ollivier and Villani's inequality is optimal, we believe that the curvature term in our inequality can be improved. We identify a hypothetical concentration inequality in the symmetric group and prove that it yields an optimally curved Brunn-Minkowski inequality.

math.CO↗