Chi Hoi Yip

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 the classical work on entrywise preservers by 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 , 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 case where the sign preservers are extensions of power functions. These results are built on our classification of entrywise positivity preservers over broader complex domains. Our results yield a complementary connection with recent work of Belton, Guillot, Khare, and Putinar 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.

We investigate ‐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 ‐Diophantine ‐tuples, answering a question of Hammonds et al., and study some related questions for ‐Diophantine sets, extending and improving several recent works. We also sharpen a classical estimate of Chung and Graham on even partial octahedra in Paley sum hypergraphs.

Let and . A Diophantine tuple with property is a set of positive integers such that is a th power for all with . 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.

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.