On Proinov's lower bound for the diaphony

TL;DR

This paper revisits Proinov's 1986 lower bound for diaphony, providing an English proof, improving constants, and extending the results to dyadic diaphony for sequences.

Contribution

It offers a self-contained English proof of Proinov's lower bound and extends the results to dyadic diaphony with improved asymptotic constants.

Findings

Provided an English version of Proinov's proof.

Improved asymptotic constants using recent results.

Derived explicit lower bounds for dyadic diaphony.

Abstract

In 1986, Proinov published an explicit lower bound for the diaphony of both finite and infinite sequences of points in the d-dimensional unit cube. To the best of our knowledge, the proofs of these results were so far only available in a monograph of Proinov written in Bulgarian. The first contribution of our paper is to give a self contained version of Proinov's proof in English. Along the way, we improve the asymptotic constants using recent and corrected results of Hinrichs and Markhasin. Finally, as a main result, we use the method of Proinov to derive an explicit lower bound for the dyadic diaphony of finite and infinite sequences in a similar fashion.