TY - JOUR T1 - On Generalizations of $p$-Sets and Their Applications AU - Heng Zhou & Zhiqiang Xu JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 453 EP - 466 PY - 2018 DA - 2018/12 SN - 12 DO - http://doi.org/10.4208/nmtma.OA-2017-0145 UR - https://global-sci.org/intro/article_detail/nmtma/12904.html KW - AB -

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