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

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

Published online: 2018-12

Preview Full PDF 116 1464
Export citation
• 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.

• Keywords

@Article{NMTMA-12-453, author = {Heng Zhou and Zhiqiang Xu}, title = {On Generalizations of $p$-Sets and their Applications}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2018}, volume = {12}, number = {2}, pages = {453--466}, 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.