The $\boldsymbol{p}$-adic diaphony of the Halton sequence

TL;DR

This paper links the p-adic diaphony, a measure of distribution irregularity, to worst-case integration error in a specific Hilbert space, providing bounds for the Halton sequence's diaphony.

Contribution

It interprets p-adic diaphony as a worst-case error in a reproducing kernel Hilbert space and establishes an upper bound for the Halton sequence.

Findings

Upper bound on the p-adic diaphony of the Halton sequence

Interpretation of diaphony as worst-case integration error

Connection between distribution irregularity and Hilbert space error

Abstract

The $\boldsymbol{p}$-adic diaphony as introduced by Hellekalek is a quantitative measure for the irregularity of distribution of a sequence in the unit cube. In this paper we show how this notion of diaphony can be interpreted as worst-case integration error in a certain reproducing kernel Hilbert space. Our main result is an upper bound on the $\boldsymbol{p}$-adic diaphony of the Halton sequence.