arXiv subjects

Pitchayut Saengrungkongka

Publications and source records attributed to Pitchayut Saengrungkongka.

6 recordsLinked to original sources

arXiv2026

Sparse $k$-AP Covering Sets and the Arithmetic Kakeya Conjecture

A subset $A\subseteq \mathbb N_0$ is $k$-AP covering if there exists a constant $n_0$ such that for every integer $x>n_0$, there exists $d\in\mathbb N_0$ such that $x-d, x-2d,\dots,x-(k-1)d$ are all in $A$. Disproving a conjecture of Kiss, Sándor, and Yang, we prove that for every integer $k\geq 6$, there exists a constant $\varepsilon=\varepsilon_k>0$ and a $k$-AP covering set $A$ such that $|A\cap \{0,1,\dots,n\}| < n^{\frac{k-2}{k-1}-\varepsilon}$ for all sufficiently large $n$. We also relate this problem to the Arithmetic Kakeya Conjecture by Katz and Tao.

math.CO↗

arXiv2026

Improved Bounds on Rainbow $k$-partite Matchings

Let $n$, $s$, and $k$ be positive integers. We say that a sequence $f_1,\dots,f_s$ of nonnegative integers is satisfying if for any collection of $s$ families $\mathcal F_1,\dots,\mathcal F_s\subseteq [n]^k$ such that $|\mathcal F_i|=f_i$ for all $i$, there exists a rainbow matching, i.e., a list of pairwise disjoint tuples $F_1\in\mathcal F_1$, $\dots$, $F_s\in\mathcal F_s$. We investigate the question, posed by Kupavskii and Popova, of determining the smallest $c=c(n,s,k)$ such that the arithmetic progression $c$, $n^{k-1}+c$, $2n^{k-1}+c$, $\dots$, $(s-1)n^{k-1}+c$ is satisfying. We prove that the sequence is satisfying for $c=Ω_k(\max(s^2n^{k-2}, sn^{k-3/2}\sqrt{\log s}))$, improving the previous result by Kupavskii and Popova. We also study satisfying sequences for $k=2$ using the polynomial method, extending the previous result by Kupavskii and Popova to when $n$ is not prime.

math.CO↗

arXiv2026

Extremal $t$-intersecting Families of Permutations for Large $t$

A set of permutations of $\{1,2,\dots,n\}$ is $t$-intersecting if any two permutations agree on at least $t$ inputs. A recent work by Kupavskii, in the spirit of the Erdős-Ko-Rado Theorem, shows that for all $t\leq n-O\left(\frac{n\log\log n}{\log n}\right)$, every $t$-intersecting family of permutations of $\{1,2,\dots,n\}$ with the maximum size must be isomorphic to the set $$A_k = \{σ: σ(i)=i\text{ for at least } t+k \text{ indices } i\in\{1,2,\dots,t+2k\}\}$$ for some $k$. By refining Kupavskii's spread approximation technique, we prove that this conclusion holds for a wider range of $t\leq n-n^{5/7+\varepsilon}$.

math.CO↗

arXiv2025

Gluing Genus 1 and Genus 2 Curves Along $\ell$-torsion

Let $Y$ be a genus $2$ curve over $\mathbb Q$. We provide a method to systematically search for possible candidates of a prime $\ell\geq 3$ and a genus $1$ curve $X$ for which there exists a genus $3$ curve $Z$ over $\mathbb Q$ whose Jacobian is, up to quadratic twist, $(\ell, \ell, \ell)$-isogenous to the product of Jacobians of $X$ and $Y$, building on the work by Hanselman, Schiavone, and Sijsling for $\ell=2$. We find several such pairs $(X,Y)$ for prime $\ell$ up to $13$. We also improve their numerical gluing algorithm, allowing us to successfully glue genus $1$ and genus $2$ curves along their $13$-torsion.

math.NT↗

arXiv2025

On $t$-intersecting Families of Spanning Trees

We prove that there exists a constant $c>0$ such that for all integers $2\leq t\leq cn$, if $\calA$ is a collection of spanning trees in $K_n$ such that any two intersect at at least $t$ edges, then $|\calA|\leq 2^tn^{n-t-2}$. This bound is tight; the equality is achieved when $\calA$ is a collection of spanning trees containing a fixed $t$ disjoint edges. This is an improvement of a result by Frankl, Hurlbert, Ihringer, Kupavskii, Lindzey, Meagher, and Pantagi, who proved such a result for $t=O\left(\frac n{\log n}\right)$.

math.CO↗

arXiv2024

Complexity of 2D Snake Cube Puzzles

Given a chain of $HW$ cubes where each cube is marked "turn $90^\circ$" or "go straight", when can it fold into a $1 \times H \times W$ rectangular box? We prove several variants of this (still) open problem NP-hard: (1) allowing some cubes to be wildcard (can turn or go straight); (2) allowing a larger box with empty spaces (simplifying a proof from CCCG 2022); (3) growing the box (and the number of cubes) to $2 \times H \times W$ (improving a prior 3D result from height $8$ to $2$); (4) with hexagonal prisms rather than cubes, each specified as going straight, turning $60^\circ$, or turning $120^\circ$; and (5) allowing the cubes to be encoded implicitly to compress exponentially large repetitions.

cs.CC↗