• Corpus ID: 118231813

Uniform distribution of sequences

  title={Uniform distribution of sequences},
  author={Lauwerens Kuipers},
( 1 ) {xn}z= Xn--Z_I Zin-Ztn-I is uniformly distributed mod 1, i.e., if ( 2 ) lim (1/N)A(x, N, {xn}z)-x (0x<_ 1), where A(x, N, {Xn)) denotes the number of indices n, l<=n<=N such that {x} is less than x. The ollowing distribution properties of the sequence (Xn)=(nO)(t an arbitrary positive real number) are well-known" (i) I Z--Zn_c and Z/Zn_xol as n-c, then (Xn) is uniformly distributed mod z (W. J. Le Veque [6]). (ii) If z-z_x is decreasing, then (Xn)is uniformly distributed modz for almost… 

Two Distribution Problems for Polynomials

Let (x n ), n = l, 2,… be a sequence of real numbers and denote by A(x n , N, x) the number of n ≤ N such that the fractional part {x n } is contained in the given interval [0, x) ⊆ [0,1). The


Let xn and yn, n = 1, 2, . . . , be sequences in the unit interval [0, 1) and let F (x, y) be a continuous function defined on [0, 1]2. In this paper we consider limit points of sequence 1 N ∑N n=1 F

A uniform distribution question related to numerical analysis

Using the theory of uniform distribution modulo one, it is shown that under certain conditions on the real-valued functions ca(x) and g(x) on [0, 1 ], -1 h fh% {hl(yh)}lmg(,yh) =(m + 1) f1f g(x) dx

Asymptotic Distribution mod m and Independence of Sequences of Integers. II

Let m_>2 be a fixed modulus. Let (an), n----i, 2,..., be a given sequence of integers. For integers N_>I and ], let A(N;], a) be the number of n, 1_n_<N, with a-2" (mod m). If a(]) lim A(N ], an) /N

The Uniform Distribution of Sequences Generated by Iterated Polynomials

  • Emil Lerner
  • Mathematics
    p-Adic Numbers, Ultrametric Analysis and Applications
  • 2019
In the paper we show that given a polynomial f over ℤ = 0, ±1, ±2, ..., deg f ⩾ 2, the sequence x, f(x), f(f(x)) = f(2)(x), ..., where x is m-adic integer, produces a uniformly distributed set of

Uniform distribution of some special sequences

We know that (f(p)), where p is n-th prime number, is uniformly distributed mod 1 if f(x) is a polynomial with real coefficients and at least one of the coefficients of f(x)--f(O) is irrational [6 or

On strong uniform distribution

Let a= (ai)i=1 be a strictly increasing sequence of natural numbers and let be a space of Lebesgue measurable functions defined on [0,1). Let {y} denote the fractional part of the real number y. We


is called the discrepancy of (xn)n=1. In 1954 Roth (see [DrTi], [KN]) proved that for any sequence in [0, 1) limN→∞ND(N)/ log N > 0. (2) Let A be an s × s invertible matrix with integer entries. A

Diophantine approximation generalized

In this paper we study the set of x ∈ [0, 1] for which the inequality |x − xn| < zn holds for infinitely many n = 1, 2, .... Here xn ∈ [0, 1) and zn s> 0, zn → 0, are sequences. In the first part of

Good points for diophantine approximation

Given a sequence (xn)n=1∞ of real numbers in the interval [0, 1) and a sequence (δn)n=1∞ of positive numbers tending to zero, we consider the size of the set of numbers in [0, 1] which can be ‘well



On uniform distribution modulo a subdivision

so that 0 < (x)^ < l Let {x̂ j be an increasing sequence of positive numbers. If the sequence ί ( ^ K l is uniformly distributed over [0, 1], in the sense that the proportion of the numbers \ ^ 1 ) Δ