Volume 12, Issue 2
On Generalizations of $p$-Sets and their Applications

Heng Zhou and Zhiqiang Xu

10.4208/nmtma.OA-2017-0145

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 453-466.

Full Article 113 169
  • 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 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.


  • History

Published online: 2018-12

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