arXiv subjects

Chi Hoi Yip

Publications and source records attributed to Chi Hoi Yip.

At least 37 records · Page 2Linked to original sources

arXiv2026

Entrywise transforms preserving matrix positivity and non-positivity

We characterize real and complex functions which, when applied entrywise to square matrices, yield a positive definite matrix if and only if the original matrix is positive definite. We refer to these transformations as sign preservers. Compared to classical work on entrywise preservers of Schoenberg and others, we completely resolve this problem in the harder fixed dimensional setting, extending a similar recent classification of sign preservers obtained for matrices over finite fields. When the matrix dimension is fixed and at least $3$, we show that the sign preservers are precisely the positive scalar multiples of the continuous automorphisms of the underlying field. This is in contrast to the $2 \times 2$ case where the sign preservers are extensions of power functions. These results are built on our classification of $2 \times 2$ entrywise positivity preservers over broader complex domains. Our results yield a complementary connection with a work of Belton, Guillot, Khare, and Putinar (2023) on negativity-preserving transforms. We also extend our sign preserver results to matrices with a structure of zeros, as studied by Guillot, Khare, and Rajaratnam for the entrywise positivity preserver problem. Finally, in the spirit of sign preservers, we address a natural extension to monotone maps, classically studied by Loewner and many others.

math.CA↗

arXiv2026

An Erdős-Ko-Rado theorem for binary codes

We study intersecting families of words from the Erdős-Ko-Rado perspective. When the alphabet size is $2$, a maximum intersecting family is not necessarily a star. However, we prove that every maximum $3$-wise intersecting family is a star. We also present a new proof of the known result for alphabets of size at least $3$: maximum intersecting families of words are exactly the stars.

math.CO↗

arXiv2026

Hypersurfaces passing through the Galois orbit of a point

Asgarli, Ghioca, and Reichstein proved that if $K$ is a field with $|K|>2$, then for any positive integers $d$ and $n$, and separable field extension $L/K$ with degree $m=\binom{n+d}{d}$, there exists a point $P\in \mathbb{P}^n(L)$ which does not lie on any degree $d$ hypersurface defined over $K$. They asked whether the result holds when $|K| = 2$. We answer their question in the affirmative by combining various ideas from arithmetic geometry. More generally, we show that for each positive integer $r$ and separable field extension $L/K$ with degree $r$, there exists a point $P \in \mathbb{P}^n(L)$ such that the vector space of degree $d$ forms over $K$ that vanish at $P$ has the expected dimension. We also discuss applications to linear systems of hypersurfaces with special properties.

math.AG↗

arXiv2026

Cliques in Paley graphs of square order and in Peisert graphs

We study maximal cliques in the collinearity graphs of Desarguesian nets, give some structural results and some numerical information. In particular, we show for Desarguesian nets that the set consisting of a point $x$ together with all its neighbors on a line $L$ (with $x$ not on $L$) is contained in a unique maximal clique $C_{x,L}$ and determine the sizes and automorphism groups of such maximal cliques $C_{x,L}$ in all cases.

math.CO↗

arXiv2026

$f$-Diophantine sets over finite fields via quasi-random hypergraphs from multivariate polynomials

We investigate $f$-Diophantine sets over finite fields via new explicit constructions of families of quasi-random hypergraphs from multivariate polynomials. In particular, our construction not only offers a systematic method for constructing quasi-random hypergraphs but also provides a unified framework for studying various hypergraphs arising from multivariate polynomials over finite fields, including Paley sum hypergraphs, and hypergraphs derived from Diophantine tuples and their generalizations. We derive an asymptotic formula for the number of $k$-Diophantine $m$-tuples, answering a question of Hammonds et al., and study some related questions for $f$-Diophantine sets, extending and improving several recent works. We also sharpen a classical estimate of Chung and Graham on even partial octahedrons in Paley sum hypergraphs.

math.CO↗

arXiv2026

On generalized covering radii of binary primitive double-error-correcting BCH codes

The generalized covering radii (GCR) of linear codes are a fundamental higher-dimensional extension of the classical covering radius. While the second and third GCR of binary primitive double-error-correcting BCH codes, $\text{BCH}(2,m)$, were recently determined, their proofs relied on highly complex combinatorial arguments, and the behavior of the GCR hierarchy for larger orders $k$ has remained largely unexplored. In this paper, we introduce the Generalized Supercode Lemma, which lower-bounds the GCR of a code using the generalized Hamming weights of an appropriate supercode. Applying this lemma, we significantly streamline and simplify the proofs for the known lower bounds of $ρ_2(\text{BCH}(2,m))$ and $ρ_3(\text{BCH}(2,m))$, and we establish a new lower bound for $ρ_4(\text{BCH}(2,m))$. Furthermore, by combining combinatorial arguments with Weil-type exponential sum estimates, we investigate the GCR hierarchy for general $k$, proving that $2k \le ρ_k(\text{BCH}(2,m)) \le 2k+1$ whenever $m$ is sufficiently large compared to $k$.

cs.IT↗

arXiv2026

Hilbert cubes in sets with arithmetic properties

In this paper, we introduce new general frameworks for estimating the maximal dimension of Hilbert cubes contained in finite truncations of arbitrary sets. As applications, we investigate Hilbert cubes in a range of arithmetic sets, including perfect powers, powerful numbers, primes, smooth numbers, and squarefree numbers. Along the way, we substantially sharpen several earlier results of Dietmann-Elshotlz, Erdős-Sárközy-Stewart, Hajdu, and Sárközy, and we obtain bounds that are sharp up to the implied constant in several cases. Additionally, we prove conditional results of independent interest, including an almost sharp uniform upper bound on the number of $k$-th powers in an arithmetic progression for each $k\geq 4$, assuming the ABC conjecture.

math.NT↗

arXiv2026

Multiplicative irreducibility of shifted multiplicative subgroups

In a recent breakthrough, Kalmynin resolved conjectures of Lev--Sonn and Sárközy on additive decompositions of multiplicative subgroups of prime fields. In this paper, inspired by a related conjecture of Sárközy, we prove multiplicative analogues of Kalmynin's results. We show that for every proper multiplicative subgroup $G$, the shifted set $(G-1)\setminus\{0\}$ cannot be written as a product set nontrivially, addressing a conjecture of Sárközy. In addition, we prove that no nonzero shift of any coset of a proper multiplicative subgroup is a ratio set of the form $A/A$. Our results substantially sharpen previous theorems of Shkredov and the authors.

math.CO↗

arXiv2026

Bipartite Diophantine tuples and their applications

This paper investigates bipartite variants of generalized Diophantine tuples and their applications. We generalize a result of Bugeaud--Dujella on a special family of bipartite Diophantine tuples and affirmatively resolve a related question posed by the second author. Additionally, we establish new connections between bipartite Diophantine tuples and several known variants of Diophantine tuples, including those introduced by Banks--Luca--Szalay and Kihel--Kihel.

math.NT↗

arXiv2026

Diophantine tuples and product sets in shifted powers

Let $k\geq 2$ and $n\neq 0$. A Diophantine tuple with property $D_k(n)$ is a set of positive integers $A$ such that $ab+n$ is a $k$-th power for all $a,b\in A$ with $a\neq b$. Such generalizations of classical Diophantine tuples have been studied extensively. In this paper, we prove several results related to robust versions of such Diophantine tuples and discuss their applications to product sets contained in a nontrivial shift of the set of all perfect powers or some of its special subsets. In particular, we substantially improve several results by Bérczes--Dujella--Hajdu--Luca, and Yip. We also prove several interesting conditional results. Our proofs are based on a novel combination of ideas from sieve methods, Diophantine approximation, and extremal graph theory.

math.NT↗

arXiv2026

Van Lint-MacWilliams' conjecture and maximum cliques in Cayley graphs over finite fields, II

The well-known Van Lint--MacWilliams' conjecture states that if $q$ is an odd prime power, and $A\subseteq \mathbb{F}_{q^2}$ such that $0,1 \in A$, $|A|=q$, and $a-b$ is a square for each $a,b \in A$, then $A$ must be the subfield $\mathbb{F}_q$. This conjecture was first proved by Blokhuis and is often phrased in terms of the maximum cliques in Paley graphs of square order. Previously, Asgarli and the author extended Blokhuis' theorem to a larger family of Cayley graphs. In this paper, we give a new simple proof of Blokhuis' theorem and its extensions. More generally, we show that if $S \subseteq \mathbb{F}_{q^2}^*$ has small multiplicative doubling, and $A\subseteq \mathbb{F}_{q^2}$ with $0,1 \in A$, $|A|=q$, such that $A-A \subseteq S \cup \{0\}$, then $A=\mathbb{F}_q$. This new result refines and extends several previous works; moreover, our new approach avoids using heavy machinery from number theory.

math.CO↗

arXiv2025

An inverse theorem on sets with rich additive structure modulo primes

In this paper, we prove several results on the structure of maximal sets $S \subseteq [N]$ such that $S$ mod $p$ is contained in a short arithmetic progression, or the union of short progressions, where $p$ ranges over a subset of primes in an interval $[y,2y]$ with $(\log N)^{O(1)} < y \leq N$. We also provide several constructions demonstrating the sharpness of our results. Furthermore, as an application, we provide several improvements on the larger sieve bound for $|S|$ when $S$ mod $p$ has strong additive structure, parallel to the work of Green--Harper and Shao for improvements on the large sieve.

math.NT↗

arXiv2025

Paley-like quasi-random graphs arising from polynomials

Paley graphs and Paley sum graphs are classical examples of quasi-random graphs. In this paper, we provide new constructions of families of quasi-random graphs that behave like Paley graphs but are neither Cayley graphs nor Cayley sum graphs. These graphs give a unified perspective of studying various graphs arising from polynomials over finite fields, such as Paley graphs, Paley sum graphs, and graphs arising from Diophantine tuples and their generalizations. We also obtain lower bounds on the clique and independence numbers of the graphs in these families.

math.CO↗

arXiv2025

Multiplicative irreducibility of small perturbations of the set of shifted $k$-th powers

Motivated by a conjecture of Erdős on the additive irreducibility of small perturbations of the set of squares, recently Hajdu and Sárközy studied a multiplicative analogue of the conjecture for shifted $k$-th powers. They conjectured that for each $k\geq 2$, if one changes $o(X^{1/k})$ elements of $M_k'=\{x^k+1: x \in \mathbb{N}\}$ up to $X$, then the resulting set cannot be written as a product set $AB$ nontrivially. In this paper, we confirm a more general version of their conjecture for $k\geq 3$.

math.NT↗

arXiv2025

Intersective sets over abelian groups

Given a finite abelian group $G$ and a subset $J\subset G$ with $0\in J$, let $D_{G}(J,N)$ be the maximum size of $A\subset G^{N}$ such that the difference set $A-A$ and $J^{N}$ have no non-trivial intersection. Recently, this extremal problem has been widely studied for different groups $G$ and subsets $J$. In this paper, we generalize and improve the relevant results by Alon and by Hegedűs by building a bridge between this problem and cyclotomic polynomials with the help of algebraic graph theory. In particular, we construct infinitely many non-trivial families of $G$ and $J$ for which the current known upper bounds on $D_{G}(J, N)$ can be improved exponentially.

math.CO↗

arXiv2025

Generalized additive bases and difference bases for Cartesian product of finite abelian groups

For a finite group $G$ and positive integer $g$, a $g$-additive basis is a subset of $G$ whose pairwise sums cover each element of $G$ at least $g$ times, with $g$-difference bases defined similarly using pairwise differences. While prior work focused on $1$-additive and $1$-difference bases, recent works of Kravitz and Schmutz--Tait explored $g$-additive and $g$-difference bases in finite abelian groups. This paper investigates such bases in $G^n$, the Cartesian product of a finite abelian group $G$. We construct $g$-additive and $g$-difference bases in $G^n$, which lead to asymptotically sharp upper bounds on the minimal sizes of such bases. Our proofs draw on ideas from additive combinatorics and combinatorial design theory.

math.CO↗

arXiv2025

Oscillation results for the summatory functions of fake mu's

Mossinghoff, Trudgian, and the first author~\cite{MMT23} recently introduced a family of arithmetic functions called ``fake $μ$'s'', which are multiplicative functions for which there is a $\{-1,0,1\}$-valued sequence $(\varepsilon_j)_{j=1}^{\infty}$ such that $f(p^j) = \varepsilon_j$ for all primes $p$. They investigated comparative number-theoretic results for fake $μ$'s and in particular proved oscillation results at scale $\sqrt{x}$ for the summatory functions of fake $μ$'s with $\varepsilon_1=-1$ and $\varepsilon_2=1$. In this paper, we establish new oscillation results for the summatory functions of all nontrivial fake $μ$'s at scales $x^{1/2\ell}$ where $\ell$ is a positive integer (the ``critical index'') depending on $f$; for $\ell=1$ this recovers the oscillation results in~\cite{MMT23}. Our work also recovers results on the indicator functions of powerfree and powerfull numbers; we generalize techniques applied to each of these examples to extend to all fake $μ$'s.

math.NT↗

arXiv2025

Exact values and improved bounds on the clique number of cyclotomic graphs

Let $q$ be an odd power of a prime $p$, and $S \subset \mathbb{F}_q^*$ such that $S=-S$ and $S/S \neq \mathbb{F}_q^*$. We show that the clique number of the Cayley graph $\operatorname{Cay}(\mathbb{F}_q^+,S)$ is at most $\sqrt{|S/S|}+\sqrt{q/p}$, improving the best-known $\sqrt{q}$ upper bound for many families of such graphs substantially. Such a new bound is strongest for cyclotomic graphs and in particular, it implies the first nontrivial upper bound on the clique number of all generalized Paley graphs of non-square order, extending the work of Hanson and Pertidis. Moreover, our new bound is asymptotically sharp for an infinite family of generalized Paley graphs, and we further discover the first nontrivial family among them for which the clique number can be exactly determined. We also obtain a new lower bound on the number of directions determined by a large Cartesian product in the affine Galois plane $AG(2,q)$, which is sharp for infinite families.

math.CO↗