On Generalizations of $p$-Sets and their Applications

The $p$-set, which is in a simple analytic form, is well distributed in unit cubes. The well-known Weilu0027s 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.