One of the most beautiful, and useful, areas in which quasirandomness has been studied concerns the subsets of the (rational) integers modulo n, which we write Zn. We need a few items of notation before introducing the random-like properties we are interested in. First of all, for x ∈ Zn, let en(x) = e. When n is understood, we simply write e(x). Then, for a subset S ⊂ Zn, let the function χS :...

A primary example is to take an integer g ≥ 2 and set a(x) = g. As is true for any increasing sequence of integers, for almost every α the fractional parts αa(x) are uniformly distributed modulo 1. Moreover, for lacunary sequences, it has long been known that the fractional parts of αa(x) have strong randomness properties. For instance, the exponential sums 1 √ N ∑ x≤N cos(2παa(x)) have a Gauss...

Abstract: In this paper, we consider the coordinated attitude control problem without velocity measurements. Based on the recently introduced unit quaternion output feedback for the attitude tracking of a rigid body, we present a class of decentralized coordinated control laws to solve the alignment problem for a group of spacecraft within a formation without velocity measurements. The approach...

Let b 2 be an integer and wb(n) be the sum of digits of the nonnegative integer n written in hereditary base b notation. We give optimal upper bounds for the exponential sum PN 1 n=0 exp(2⇡iwb(n)t), where t is a real number. In particular, our results imply that for each positive integer m the sequence {wb(n)}n=0 is uniformly distributed modulo m; and that for each irrational real ↵ the sequenc...

We discuss the distinctness problem of the reductions modulo M of maximal length sequences modulo powers of an odd prime p, where the integer M has a prime factor different from p. For any two different maximal length sequences generated by the same polynomial, we prove that their reductions modulo M are distinct. In other words, the reduction modulo M of a maximal length sequence is proved to ...

It was rediscovered several times, can be constructed in many alternative ways and occurs in various fields of mathematics (see the survey [1]). The set of all overlap-free words was studied e. g. by Fife [8] who described all binary overlap-free infinite words and Séébold [13] who proved that the Thue-Morse word is essentially the only binary overlap-free word which is a fixed point of a morph...

is the ordinary formal power series generating function for the sequence {yn+i}„>0 (cf. [12]. Furthermore, it is easy to see [1] that when the discriminant A = X +4ju ofP(t) is nonnegative and X & 0, the ratios yn+l I yn converge (in the usual archimedean metric on U) to a reciprocal root a of P(t). In this article we show that ratios of these y n also exhibit rapid convergence properties relat...

Sampling random graphs with given properties is a key step in the analysis of networks, as random ensembles represent basic null models required to identify patterns such as communities and motifs. A key requirement is that the sampling process is unbiased and efficient. The main approaches are microcanonical, i.e. they sample graphs that exactly match the enforced constraints. Unfortunately, w...