On Proinov’s Lower Bound for the Diaphony

In 1986, Proinov published an explicit lower bound for the diaphony of finite and infinite sequences of points contained in the d−dimensional unit cube [Proinov, P. D.:On irregularities of distribution, C. R. Acad. Bulgare Sci. 39 (1986), no. 9, 31–34]. However, his widely cited paper does not contain the proof of this result but simply states that this will appear elsewhere. To the best of our knowledge, this proof was so far only available in a monograph of Proinov written in Bulgarian [Proinov, P. D.: Quantitative Theory of Uniform Distribution and Integral Approximation, University of Plovdiv, Bulgaria (2000)]. 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 explicit asymptotic constants implementing recent, and corrected results of [Hinrichs, A.—Markhasin, L.: On lower bounds for the ℒ2-discrepancy, J. Complexity 27 (2011), 127–132.] and [Hinrichs, A.—Larcher, G.: An improved lower bound for the ℒ2-discrepancy, J. Complexity 34 (2016), 68–77]. (The corrections are due to a note in [Hinrichs, A.—Larcher, G. An improved lower bound for the ℒ2-discrepancy, J. Complexity 34 (2016), 68–77].) 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.

On Irregularities of Distribution of Binary Sequences Relative to Arithmetic Progressions, II (Constructive Bounds)

In Part I of this paper we studied the irregularities of distribution of binary sequences relative to short arithmetic progressions. First we introduced a quantitative measure for this property. Then we studied the typical and minimal values of this measure for binary sequences of a given length. In this paper our goal is to give constructive bounds for these minimal values.

Additive Energy and Irregularities of Distribution

We consider strictly increasing sequences (an)n≥1 of integers and sequences of fractional parts ({anα})n≥1 where α ∈ R. We show that a small additive energy of (an)n≥1 implies that for almost all α the sequence ({anα})n≥1 has large discrepancy. We prove a general result, provide various examples, and show that the converse assertion is not necessarily true.

On Irregularities of Distribution of Binary Sequences Relative to Arithmetic Progressions, I. (General Results)

In 1964 K. F. Roth initiated the study of irregularities of distribution of binary sequences relative to arithmetic progressions and since that numerous papers have been written on this subject. In the applications one needs binary sequences which are well distributed relative to arithmetic progressions, in particular, in cryptography one needs binary sequences whose short subsequences are also well-distributed relative to arithmetic progressions. Thus we introduce weighted measures of pseudorandomness of binary sequences to study this property. We study the typical and minimal values of this measure for binary sequences of a given length.

Klaus Friedrich Roth. 29 October 1925 — 10 November 2015

Klaus Friedrich Roth, who died in Inverness on 10 November 2015 aged 90, made fundamental contributions to different areas of number theory, including diophantine approximation, the large sieve, irregularities of distribution and what is nowadays known as arithmetic combinatorics. He was the first British winner of the Fields Medal, awarded in 1958 for his solution in 1955 of the famous Siegel conjecture concerning approximation of algebraic numbers by rationals. He was elected a Fellow of the Royal Society in 1960, and received its Sylvester Medal in 1991. He was also awarded the De Morgan Medal of the London Mathematical Society in 1983, and elected Fellow of University College London in 1979, Honorary Fellow of Peterhouse in 1989, Honorary Fellow of the Royal Society of Edinburgh in 1993 and Fellow of Imperial College London in 1999.

SOLUTIONS TO POLYNOMIAL CONGRUENCES IN WELL-SHAPED SETS

We use a generalisation of Vinogradov’s mean value theorem of Parsell et al. [‘Near-optimal mean value estimates for multidimensional Weyl sums’, arXiv:1205.6331] and ideas of Schmidt [‘Irregularities of distribution. IX’, Acta Arith. 27 (1975), 385–396] to give nontrivial bounds for the number of solutions to polynomial congruences, when the solutions lie in a very general class of sets, including all convex sets.