On generalizations of $p$-sets and their applications
For researchers in numerical analysis and signal processing, this offers more flexible point sets, but the reliance on an unproven conjecture limits practical impact.
The paper generalizes $p$-sets to more flexible constructions, and under the Goldbach conjecture, produces point sets of arbitrary even cardinality, overcoming a key limitation of original $p$-sets. Upper bounds on exponential sums are provided, suggesting potential applications in numerical integration, compressed sensing, and UQ.
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 $\mathcal{P}_{d,p}^{{\mathbf a},ε}$, 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 $\mathcal{P}_{d,p}^{{\mathbf a},ε}$ and ${\mathcal L}_{p,q}$, which imply these sets have many potential applications.