Combinatorial designs and the Prouhet--Tarry--Escott problem

Abstract

This is the first paper that provides a systematic treatment of the $r$-dimensional PTE problem in additive number theory, abbreviated by PTE$_r$, through its connection with combinatorial design theory, the branch of combinatorial mathematics that deals with finite set systems or arrangements with the ^^ balancedness' conditions. We first propose a combinatorial reconsideration of the definition of nontrivial solution introduced by Alpers and Tijdeman (2007), and then prove a fundamental lower bound for the size of such solutions. We exhibit high-dimensional minimal solutions with respect to the fundamental bound, which inherently have the structure of distinctive block designs or orthogonal arrays (OAs). Next, we develop a powerful method for constructing PTE$_r$ solutions via various classes of combinatorial designs such as block designs and OAs. Furthermore, we explore two dimension-lifting methods for constructing PTE$_r$ solutions: one is a combinatorial composition that produces PTE$_r$ solutions by embedding lower-dimensional solutions into OAs with $r$ columns, and the other is a recursive technique in which a PET$_r$ solution is constructed by taking the Cartesian product of two lower-dimensional solutions. It is emphasized that our results generalize many previous works, including a measure-theoretic construction by Lorentz (1949) and its geometric analog by Alpers and Tijdeman (2007), a key lemma in Jacroux's work (1995) on the construction of sets of integers with equal power sums, and the famous Borwein solution and its two-dimensional extension by Matsumura and Sawa (2025). In addition, we prove a characterization theorem for ideal solutions of the PTE$_1$ and discuss the connection with a curious phenomenon, called half-integer design, that is rarely reported in the combinatorial design theory or spherical design theory.