On generalizations of $p$-sets and their applications

Abstract: The p-set, which is in a simple analytic form, is well distributed in unit cubes. The well-known Weil’s exponential sum theorem presents an upper bound of the exponential sum over the p-set. Based on the result, one shows that the p-set performs well in numerical integration, in compressed sensing as well as in UQ. However, p-set is somewhat rigid since the cardinality of the p-set is a prime p and the set only depends on the prime number p. The purpose of this paper is to present generalizations of p-sets, say P d,p, which is more flexible. Particularly, when a prime number p is given, we have many different choices of the new p-sets. Under the assumption that Goldbach conjecture holds, for any even number m, we present a point set, say Lp,q, with cardinality m− 1 by combining two different new p-sets, which overcomes a major bottleneck of the p-set. We also present the upper bounds of the exponential sums over P d,p and Lp,q, which imply these sets have many potential applications.