Equidistribution Research Articles

The Size of Trigonometric and Walsh Series and Uniform Distribution Mod 1

We characterize the class of non-decreasing functions f such that for any increasing sequence {nk} of integers lim N → ∞ Σ k ⩽ N cos 2 π n k ω N 1 / 2 f ( N ) = 0 a.e. Combined with an inequality of Koksma our results prove the existence of an increasing sequence {nk} of integers such that the discrepancy DN(ω) of the sequence {nkω} mod 1 satisfies lim sup N → ∞ ( N / log N ) 1 / 2 D N ( ω ) = ∞ a.e. This disproves conjectures of Erdős [9] and R. C. Baker [2]. We prove the analogue of the above result for the Walsh system, thereby solving a problem of Révész [18]. Finally we solve a problem raised in [16], by showing the existence of sequences {nk} of integers with n k + 1 n k ⩾ 1 + ρ k for k ⩾ 1 , where {ρk} is a given non-increasing sequence of real numbers, for which the discrepancy DN(ω) of {nkω} mod 1 fails the upper half of the law of the iterated logarithm by a factor of log log (1/ρN).

On the asymptotically uniform distribution modulo 1 of extreme order statistics

Let (Xm)∞1 be a sequence of independent and identically distributed random variables. We give sufficient conditions for the fractional part of rnax (X1., Xn) to converge in distribution, as n←∞ to a random variable with a uniform distribution on [0, 1).

Equidistribution and the Riemann hypothesis

In this paper we demonstrate the relationship between uniform distribution modulo 1 of the sequence $cp^{\alpha}$, $p$ prime, and the zero free regions of the Riemann zeta function.

Walsh functions and uniform distribution mod $1$

Walsh functions and uniform distribution mod $1$

A remark on random and equidistributed sequences

The average of the values of a function f on the points of an equidistributed sequence in [0, 1]s converges to the integral of f as soon as f is Riemann integrable. Some known low discrepancy sequences perform faster integration than independent random sampling (cf. [1]). We show that a small random absolutely continuous perturbation of an equidistributed sequence allows to integrate bounded Borel functions, and more generally that, if the law of the random perturbation doesn't charge polar sets, such perturbed sequences allow to integrate bounded quasi-continuous functions.

On the uniform distribution modulo one of some log-like sequences

On the uniform distribution modulo one of some log-like sequences

Notes on uniform distribution modulo one

AbstractWe 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.

Limitations to the Equi-Distribution of Primes I

Limitations to the Equi-Distribution of Primes I

On the distribution of sequences connected with digit-representation

Let N=e(s)b(s)+e(s−1)b(s−1)+...+e(1)b(1)+e(0) be the digit representation of the integer N to base b or to α-scale, that is with respect to the best approximation denominators of an irrational number α. Let f:N0 → Z with f(0)=0 be an arbitrary function and r(0),r(1),... be an arbitrary sequence of integers and F(N):=f(e(s))r(s)+...+f(e (1))r(1)+f(e(0))r(0). Conditions for the uniform distribution modulo one of the sequence {F(N)x}N∈N, x∈ℝ are given.

Trigonometric approximation and uniform distribution modulo one

We construct n n -dimensional versions of the Beurling and Selberg majorizing and minorizing functions and use them to prove results on trigonometric approximation and to prove an n n -dimensional version of the Erdös-Turán inequality. Finally, an application is given to counting solutions of polynomial congruences.

Vacuum states and equidistribution of the random sequence for Glimm's scheme

Etude des schemas aux differences de Glimm engendres par des suites equireparties, appliques au probleme a valeurs initiales pour les equations de la dynamique des gaz isentropiques. Absence de vide dans l'ecoulement et leurs approximations par des differences de Glimm constituees d'onde de rarefaction

On the discrepancy of the sequence formed from multiples of an irrational number

This paper demonstrates a connection between two measures of discrepancy of sequences which arise in the theory of uniform distribution modulo one. The sequence formed from the non-negative integer multiples of an irrational number ξ is investigated and, by an application of the “Steinhaus Conjecture”, some values of the two discrepancies are obtained using continued fractions.

Generalization of two results of the theory of uniform distribution

For a sequence x 1 , … , x N {x_1}, \ldots ,{x_N} of points in [ 0 , 1 ] [0,1] and a sequence p 1 , … , p N ( p 1 + p 2 + ⋯ + p N = 1 ) {p_1}, \ldots ,{p_N}({p_1} + {p_2} + \cdots + {p_N} = 1) of nonnegative numbers, define the distribution function \[ g ( x ) = x − ∑ x k > x p k . g(x) = x - \sum \limits _{{x_k} > x} {{p_k}.} \] Let φ \varphi be an increasing function on [ 0 , 1 ] [0,1] and φ ( 0 ) = 0 \varphi (0) = 0 . The main result of the paper is \[ F ( D N ) ⩽ ∫ 0 1 φ ( | g ( x ) | ) d x ⩽ φ ( D N ) , F({D_N}) \leqslant \int _0^1 {\varphi \left ( {\left | {g(x)} \right |} \right )} dx \leqslant \varphi ({D_N}), \] where D N {D_N} is the supremum norm of g g on [ 0 , 1 ] [0,1] and F F is the antiderivative of φ \varphi with F ( 0 ) = 0 F(0) = 0 . This result generalizes and improves an estimate of Niederreiter [1] for the L 2 {L^2} discrepancy of the sequence x 1 , … , x N {x_1}, \ldots ,{x_N} . Applying the above inequality we also obtain a new criterion for uniform distribution modulo one.

The asymptotic uniform distribution modulo 1 of cumulative processes

It is proved that strongly cumulative processes (Y(t)t≧0) are asymptotically uniformly distributed modulo 1, i.e. for 0≦x<1. Thus a result due to Loynes [4] is generalized.

$\alpha$-additive functions and uniform distribution modulo one

$\alpha$-additive functions and uniform distribution modulo one

Dichteschätzungen mit gleichverteilungsmethoden

Density estimates with methods of uniform distribution mod 1. Some nonparametric multivariate density estimators for continuous functions, based on the Fejer and Jackson kernel, are presented. Uniform strong consistency results are obtained with methods of uniform distribution mod 1.

On the uniform distribution modulo one of subsequences of polynomial sequences II

Let P be a polynomial. As in the article [ J. Coquet, J. Number Theory 10 (1978) , 291–296], we find a necessary and sufficient condition for some subsequences of ( P( n)) to be uniformly distributed modulo one. These subsequences are defined by properties of the q-adic expansion of n.

On the uniform distribution modulo one of some subsequences of polynomial sequences

Let P be a polynomial. We find a necessary and sufficient condition for some subsequences of ( P( n)) to be uniformly distributed modulo one. These subsequences are defined by properties of the q-adic expansion of n.

A Tauberian theorem related to Weyl's criterion

Let { x n } be a sequence of real numbers and let a( n) be a sequence of positive real numbers, with A( N) = Σ n=1 N a( n). Tsuji has defined a notion of a(n)-uniform distribution mod 1 which is related to the problem of determining those real numbers t 0 for which A( N) −1 Σ n=1 N a( n) e − it 0 x n → 0 as N → ∞. In case f( s) = Σ n=1 ∞ a( n) e − sx n , s = σ + it, is analytic in the right half-plane 0 < σ, and satisfies a certain smoothness condition as σ → 0 +, we show that f( σ) −1 f( σ + it 0) → 0 as σ → 0 + if and only if A( N) −1 Σ n=1 N a( n) e − it 0 x n → 0 as N → ∞.

Letter to the Editor Generalized Fibonacci Numbers and Uniform Distribution Mod 1

Letter to the Editor Generalized Fibonacci Numbers and Uniform Distribution Mod 1