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0 - 1 sequences having the same numbers of ( 1 - 1 ) -couples of given distances

Antonín Lešanovský, Jan Rataj, Stanislav Hojek (1992)

Mathematica Bohemica

Let a be a 0 - 1 sequence with a finite number of terms equal to 1. The distance sequence δ ( a ) of a is defined as a sequence of the numbers of ( 1 - 1 ) -couples of given distances. The paper investigates such pairs of 0 - 1 sequences a , b that a is different from b and δ ( a ) = δ ( b ) .

A generalization of a theorem of Erdős-Rényi to m-fold sums and differences

Kathryn E. Hare, Shuntaro Yamagishi (2014)

Acta Arithmetica

Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define r N ( m ) ( ω ) to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that r N ( m ) ( ω ) < K for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.

A generalization of NUT digital (0,1)-sequences and best possible lower bounds for star discrepancy

Henri Faure, Friedrich Pillichshammer (2013)

Acta Arithmetica

In uniform distribution theory, discrepancy is a quantitative measure for the irregularity of distribution of a sequence modulo one. At the moment the concept of digital (t,s)-sequences as introduced by Niederreiter provides the most powerful constructions of s-dimensional sequences with low discrepancy. In one dimension, recently Faure proved exact formulas for different notions of discrepancy for the subclass of NUT digital (0,1)-sequences. It is the aim of this paper to generalize the concept...

A sequence adapted from the movement of the center of mass of two planets in solar system

Jana Fialová (2018)

Communications in Mathematics

In this paper we derive a sequence from a movement of center of~mass of arbitrary two planets in some solar system, where the planets circle on concentric circles in a same plane. A trajectory of center of mass of the planets is discussed. A sequence of points on the trajectory is chosen. Distances of the points to the origin are calculated and a distribution function of a sequence of the distances is found.

Almost periodic sequences and functions with given values

Michal Veselý (2011)

Archivum Mathematicum

We present a method for constructing almost periodic sequences and functions with values in a metric space. Applying this method, we find almost periodic sequences and functions with prescribed values. Especially, for any totally bounded countable set  X in a metric space, it is proved the existence of an almost periodic sequence { ψ k } k such that { ψ k ; k } = X and ψ k = ψ k + l q ( k ) , l for all  k and some q ( k ) which depends on  k .

Density of some sequences modulo 1

Artūras Dubickas (2012)

Colloquium Mathematicae

Recently, Cilleruelo, Kumchev, Luca, Rué and Shparlinski proved that for each integer a ≥ 2 the sequence of fractional parts a / n n = 1 is everywhere dense in the interval [0,1]. We prove a similar result for all Pisot numbers and Salem numbers α and show that for each c > 0 and each sufficiently large N, every subinterval of [0,1] of length c N - 0 . 475 contains at least one fractional part Q(αⁿ)/n, where Q is a nonconstant polynomial in ℤ[z] and n is an integer satisfying 1 ≤ n ≤ N.

Discrépance de la suite ( { n α } ) , α = ( 1 + 5 ) / 2

Yves Dupain (1979)

Annales de l'institut Fourier

Soit D * ( N ) la discrépance “à l’origine” de la suite n 1 + 5 2 . Nous montrons que lim sup D * ( N ) Log N = 3 20 Log 1 + 5 2 - 1 = 0 . 31 , quantité inférieure à celle correspondant à la suite de van der Corput. Les techniques utilisées sont celles liées au développement en fraction continue.

Discrépance des suites de Farey

François Dress (1999)

Journal de théorie des nombres de Bordeaux

On étudie la discrépance absolue de la suite de Farey d’ordre n et on montre, en utilisant notamment une majoration d’une intégrale portant sur la fonction sommatoire de la fonction de Möbius, qu’elle est égale à 1 n exactement, ce qui est la valeur locale au point d’abscisse 1 n .

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