# On generalizations of $p$-sets and their applications

## TL;DR

This paper introduces generalized $p$-sets that are more flexible than traditional $p$-sets, providing new constructions with potential applications in numerical analysis and related fields, assuming Goldbach's conjecture.

## Contribution

The paper develops generalized $p$-sets $\\mathcal{P}_{d,p}^{{\mathbf a},\epsilon}$ and constructs point sets ${\mathcal L}_{p,q}$ under Goldbach's conjecture, overcoming limitations of classical $p$-sets.

## Key findings

- Upper bounds for exponential sums over the new $p$-sets.
- Construction of point sets with arbitrary even cardinalities.
- Potential applications in numerical integration, compressed sensing, and UQ.

## 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 $\mathcal{P}_{d,p}^{{\mathbf a},\epsilon}$, 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 ${\mathcal L}_{p,q}$, 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 $\mathcal{P}_{d,p}^{{\mathbf a},\epsilon}$ and ${\mathcal L}_{p,q}$, which imply these sets have many potential applications.