Title: On generalizations of $p$-sets and their applications,
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 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.