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... This kind of block sequences were studied in papers, [1] , [3] , [4] , [16] and [20]. Also other kinds of block sequences were studied by several authors, see [2], [6], [8], [12] and [19]. Let Y = (y n ) be an increasing sequence of positive integers. ...
Properties of dispersion of block sequences were investigated by J. T. Tóth, L. Mišík and F. Filip [Math. Slovaca 54, 453–464 (2004; Zbl 1108.11017)]. The present paper is a continuation of the study of relations between the density of the block sequence and so called dispersion of the block sequence.
The real sequence (x n ) is maldistributed if for any non-empty interval I, the set { n ∈ : x n ∈I} has upper asymptotic density 1. The main result of this note is that the set of all maldistributed real sequences is a residual set in the set of all real sequences (i.e., the maldistribution is a typical property in the sense of Baire categories). We also generalize this result.
In 1857, Kronecker [10] showed that if θ 1 ,…, θ n are the roots of the polynomial P(z) = z ⁿ |c ⁿ⁻¹ + … + c n , where c 1 , …, c n are integers with c n ≠0, and if |θ 1 | ≤ 1, …, |θ 1 | ≤1, then θ 1 , …, θ n are roots of unity. The proof is short and ingenious: Consider the polynomials P m (z) whose roots are The condition on the size of the roots and the fact that the c i are integers implies that there can only be a finite number of different P m . Thus two distinct powers of each root must coincide and this means that each root is a root of unity.
The papers in this volume, which were presented at the 1985 Australian Mathematical Society convention, survey recent work in Diophantine analysis. The contributors are leading mathematicians in the world, and their articles are state of the art accounts, many of which include open problems pointing the way to further research. The contributions will be of general interest to number theorists and of particular interest to workers in transcendence theory, Diophantine approximation and exponential sums.
Let R0, R1, R2,… be a nondegenerate binary linear recurrence of integers defined by Rn = ARn−1 + BRn−2 (n > 1), where R0 = 0, R1 = 1, and A, B are fixed nonzero integers. It is proved that for any sufficiently large integer N ([] denotes the least common multiple of numbers).
Shift registers have been used to generate sequences of 0’s and 1’s for over thirty years. A wide variety of applications has been made of these sequences. Principally, communications have made use of the sequences generated.One particular class of shift register sequences for which applications exist is the full length nonlinear shift register sequences. These sequences are periodic and of length and all different binary n-tuples appear exactly one time in a periodic portion of the sequence. In this paper we discuss various algorithms which have been suggested for generating these sequences.
Conditions are given, weaker than those previously known, under which one can ignore a singularity while carrying out a numerical quadrature. The conditions apply in all (finite) dimensions. A special case concerns the theory of uniformly distributed sequences.
We exhibit a sequence (un) which is not uniformly distributed modulo one even though for each fixed integer k ≥ 2 the sequence (kun) is u.d. (mod 1). Within the set of all such sequences, we characterize those with a well-behaved asymptotic distribution function. We exhibit a sequence (un) which is u.d. (mod 1) even though no subsequence of the form (ukn + j) is u.d. (mod 1) for any k ≥ 2. We prove that, if the subsequences (ukn) are u.d. (mod 1) for all squarefree k which are products of primes in a fixed set P, then (un) is u.d. (mod I) if the sum of the reciprocals of the primes in P diverges. We show that this result is the best possible of its type.
Let s(d,c) be the Dedekind sum; let r be a nonzero real number. We show that the sequence of points in two-space given by 〈〉, c = 1, 2,…, 0 < d < c, (d, c) = 1, is uniformly distributed (mod 1).
Nous estimons le module des sommes trigonométriques sur la variété de dimension n – s definie par s formes en n variables, avec une forme linéaire en exposant. Cela s'applique a l'étude de la distribution des points rationnels d'une telle variété definie sur un corps fini ou sur le corps des nombres rationnels.(Received March 06 1989)
Previous investigations of this problem include those of Mordell [10], Chalk and Williams [5], and Smith [14]. Smith's main result, which encompasses the other results, can be stated as follows.(Received August 17 1980)
provided that n is odd. This is best possible for n = 3, as we shall see later. Of course we can get an exponent (1/2) + (1/(2n – 2)) trivially for even n. I do not know how to improve on this. D. R. Heath-Brown (private communication) can improve the exponent in (2) to (l/2) + ε for n ≥ 4 and prime m > C1(ε).(Received March 18 1983)
Given a subgroup G of the multiplicative group of a finite field, we investigate the number of representations of an arbitrary field element as a sum of elements, one from each coset of G. When G is of small index, the theory of cyclotomy yields exact results. For all other G, we obtain good estimates. This paper formed a portion of the author's doctoral dissertation. © 1979, University of California, Berkeley. All Rights Reserved.
