Binomial-coefficient multiples of irrationals

Cornell University - arXiv, 2017

2005

The distribution of rationals on the unit interval is filled w ith surprises. As a child, one is told that the rationals are distributed “unifo rmly” on the unit interval. If one considers the entire set Q, then yes, in a certain narrow sense, this is true. But if one considers just subsets, such as the subset of ratio nals with “small” denominators, then the distribution is far from uniform and fu ll of counter-intuitive surprises, some of which we explore below. This implies that using “intuition” to understand the rationals and, more generally, the real nu mbers is a dangerous process. Once again, we see the footprints of the set-theore tic representation of the modular groupSL(2,Z) at work. This paper is part of a set of chapters that explore the relati onship between the real numbers, the modular group, and fractals. 1 Distributions Of Rationals on the Unit Interval The entire field of classical calculus and analysis is based o n the notion that the real numbers are smoothly an...

Information and Computation

We elaborate the notions of Martin-Löf and Schnorr randomness for real numbers in terms of uniform distribution of sequences. We give a necessary condition for a real number to be Schnorr random expressed in terms of classical uniform distribution of sequences. This extends the result proved by Avigad for sequences of linear functions with integer coefficients to the wider classical class of Koksma sequences of functions. And, by requiring equidistribution with respect to every computably enumerable open set (respectively, computably enumerable open set with computable measure) in the unit interval, we give a sufficient condition for Martin-Löf (respectively Schnorr) randomness.

Functiones et Approximatio Commentarii Mathematici, 2008

Let θ be a Salem number. It is well-known that the sequence (θ n ) modulo 1 is dense but not equidistributed. In this article we discuss equidistributed subsequences. Our first approach is computational and consists in estimating the supremum of limn→∞ n/s(n) over all equidistributed subsequences (θ s(n) ). As a result, we obtain an explicit upper bound on the density of any equidistributed subsequence. Our second approach is probabilistic. Defining a measure on the family of increasing integer sequences, we show that relatively to that measure, almost no subsequence is equiditributed.

Mathematica Slovaca, 1997

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Lithuanian Mathematical Journal, 2019

Let Q 1 ,. .. , Q t ∈ R[x] be polynomials with no constant term for which each linear combination m 1 Q 1 (x) + • • • + m t Q t (x), with m 1 ,. .. , m t ∈ Z and not all 0, always has an irrational coefficient. Let I 1 ,. .. , I t be sets included in the interval [0, 1), each of which being a union of finitely many subintervals of [0, 1). Furthermore, let T be the set of those positive integers n for which {Q 1 (n)} ∈ I 1 ,. .. , {Q t (n)} ∈ I t holds simultaneously, where {y} stands for the fractional part of y. Let t 1 , t 2 ,. .. be a sequence of complex numbers uniformly summable and set T (x) = n≤x t n and T (x|T) = n≤x n∈T t n. We prove that, as x → ∞, T (x)/x ∼ T (x|T)/(λ(I 1) • • • λ(I t)x), where λ(I) stands for the Lebesgue measure of the set I.

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

We investigate the uniform distribution modulo 1 of sequence a(n) b(n) (n ∈ N), where a(n) and b(n) are suitable additive or multiplicative functions. We prove that if κ(n) is any of An r ω(n) h , An r a ω(n) , An r τ (n h) (r, h, A ∈ N) and f (n) is an arbitrary additive function, then κ(n) + f (n) is uniformly distributed mod 1. 1. Notation and introductory definitions Let A be the set of additive, M be the set of multiplicative functions. A function f : N → R belongs to A if f (nm) = f (n) + f (m) holds for every coprime pairs of integers n, m. A function g : N → R is multiplicative if g(1) = 1 and g(nm) = g(n)g(m) holds for every coprime pairs of integers n, m. Let M 1 := {g ∈ M : |g(n)| ≤ 1}.

1980

The large volume of material forces us to restrict ourselves to certain directions only. The choice of theorems and bibliographic references is entirely subjective and does not pretend to completeness. The basic problem of probabilistic number theory is the study of the distribution of values of arithmetic functions, i.e., functions defined on the set of natural numbers N. The classic investigation, taking its start in the works of K. F. Gauss and P. G. Lejeune-Dirichlet, from the point of view of the modern theory did not exceed the study of mathematical expectation in sequences of probability spaces {En, Un, ~n}, as n § Here En = [l,...,n}, U n is the set of all subsets of E n and n 1 ~ 1 A~U.. v. (A)= g m=l m~A Most arithmetic functions (a.f.) considered in number theory have the property of additivity or multiplicativity. An a.f. h(m) (respectively, g(m)) is called additive (multiplicative) if for any pair of relatively prime numbers m and n, h(m n)=h(m)+h(n) (g(m n) =g(m)g(n)). Let p be a prime. If h(p ~) =h(p) and g(pa) =g(p) for a>-l, then h(m) and g(m) are called strongly additive (s.a.) and strongly multiplicative (s.m.) a.f., respectively. Throughout the entire text following we restrict ourselves to them when this simplifies the formulation. Transition to the general case does not involve major difficulties. Starting in 1917 there have appeared papers devoted to the study of the asymptotic behavior of the statistical distribution functions ~,,(x)=v. (m<<.n, h(m)-~(n) <X) (n) where h(m) is a real a.a.f, and ~(n) and ~(n) are certain normalizing sequences. This problem occupied a central place in the problems of today. The history of the question is given in the monographs [i, 2] and in the surveys [3-5]. In papers up to the sixth decade, elementary and combinatorial methods dominated, allowing the study of separate examples of a.f. or their rather narrow classes. Only the Erdos-Wintner theorem gave an answer about the conditions on convergence of ~n(X) to the limiting function of the distribution for ~(n)= 0 and B(n)-i for any real a.a.f, h(m). 2. Kubilyus' Method In 1954-1956 I. Kubilyus developed a method allowing one to reduce the study of vn(x) to classical limit theorems of probability theory [i]. We shall explain its essence briefly. Stressing the generality of the method, we carry out the argument for a sequence of functions. This extends the class of limit laws (cf. also [7]). We represent the sequence of real s.a.a.f, hn(m) in the form h. (m) = ~] h~ (m),

Colloquium Mathematicum, 2019

In this paper we study different aspects of the Möbius randomness principle. We rephrase the Chowla, Sarnak conjectures and the Riemann hypothesis for abstract sequences and study their relationships. We, in particular, show, that in this setting the Chowla and Sarnak conjectures do not imply the Riemann hypothesis. In the second part of the paper we also study the connection between the multiplicative and additive van der Corput criteria.