@Article{NMTMA-12-2,
author = {},
title = {On Generalizations of  $p$-Sets and Their  Applications},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2019},
volume = {12},
number = {2},
pages = {453--466},
abstract = {<p style="text-align: justify;">The $p$-set, which is in a simple analytic form, is well distributed in unit cubes. The well-known Weil&#39;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&nbsp; $\mathcal{P}_{d,p}^{a,\epsilon}$ and ${\mathcal L}_{p,q}$, which imply these sets have many potential applications.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2017-0145},
url = {https://global-sci.com/article/90403/on-generalizations-of-p-sets-and-their-applications}
}