On Proinov's lower bound for the diaphony

In 1986, Proinov published an explicit lower bound for the diaphony of both finite and infinite sequences of points in the s 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 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.