Equidistribution Research Articles

The uniform distribution modulo one of certain subsequences of ordinates of zeros of the zeta function

AbstractOn the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros $1/2+i\gamma$ of the Riemann zeta function, we show that the sequence \begin{equation*}\Gamma_{[a, b]} =\Bigg\{ \gamma : \gamma>0 \quad \mbox{and} \quad \frac{ \log\big(| \zeta^{(m_{\gamma })} (\frac12+ i{\gamma }) | / (\!\log{{\gamma }} )^{m_{\gamma }}\big)}{\sqrt{\frac12\log\log {\gamma }}} \in [a, b] \Bigg\},\end{equation*} where the ${\gamma }$ are arranged in increasing order, is uniformly distributed modulo one. Here a and b are real numbers with $a<b$ , and $m_\gamma$ denotes the multiplicity of the zero $1/2+i{\gamma }$ . The same result holds when the ${\gamma }$ ’s are restricted to be the ordinates of simple zeros. With an extra hypothesis, we are also able to show an equidistribution result for the scaled numbers $\gamma (\!\log T)/2\pi$ with ${\gamma }\in \Gamma_{[a, b]}$ and $0<{\gamma }\leq T$ .

On the equidistribution of closed geodesics and geodesic nets

We show that given a closed n n -manifold M M , for a Baire-generic set of Riemannian metrics g g on M M there exists a sequence of closed geodesics that are equidistributed in M M if n = 2 n=2 ; and an equidistributed sequence of embedded stationary geodesic nets if n = 3 n=3 . One of the main tools that we use is the Weyl law for the volume spectrum for 1 1 -cycles, proved by Liokumovich, Marques, and Neves [Ann. of Math. (2) 187 (2018), pp. 933–961] for n = 2 n=2 and by Guth and Liokumovich [Preprint, arXiv:2202.11805, 2022] for n = 3 n=3 . We show that our proof of the equidistribution of stationary geodesic nets can be generalized for any dimension n ≥ 2 n\geq 2 provided the Weyl Law for 1 1 -cycles in n n -manifolds holds.

On the correlations of $n^\alpha$ mod 1

A well known result in the theory of uniform distribution modulo 1 (which goes back to Fejér and Csillag) states that the fractional parts \{n^{\alpha}\} of the sequence (n^{\alpha})_{n\ge1} are uniformly distributed in the unit interval whenever \alpha>0 is not an integer. For sharpening this knowledge to local statistics, the k -level correlation functions of the sequence (\{n^{\alpha}\})_{n\geq1} are of fundamental importance. We prove that for each k\ge2, the k -level correlation function R_{k} is Poissonian for almost every \alpha>4k^{2}-4k-1 .

Diophantine approximation with prime restriction in function fields

In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence pα when α is a fixed irrational real number and p runs over the primes. In particular, he showed that the inequality ||pα||≤p−1/5+ε has infinitely prime solutions p, where ||.|| denotes the distance to a nearest integer. This result has subsequently been improved by many authors. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality ||pα||≤p−1/3+ε. This exponent 1/3 is considered the limit of the current technology. We prove function field analogues of this result for the fields k=Fq(T) and imaginary quadratic extensions K of k. Essential in our method is the Dirichlet approximation theorem for function fields which is established in general form in the appendix authored by Arijit Ganguly.

A sharp estimate of the discrepancy of a concatenation sequence of inversive pseudorandom numbers with consecutive primes

We consider an equidistributed concatenation sequence of pseudorandom rational numbers generated from the primes by an inversive congruential method. In particular, we determine the sharp convergence rate for the star discrepancy of said sequence. Our arguments are based on well-known discrepancy estimates for inversive congruential pseudorandom numbers together with asymptotic formulae involving prime numbers.

Randomness and uniform distribution modulo one

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.

Uniform Distribution of the Weighted Sum-of-Digits Functions

Abstract The higher-dimensional generalization of the weighted q-adic sum-of-digits functions sq,γ (n), n =0, 1, 2,..., covers several important cases of sequences investigated in the theory of uniformly distributed sequences, e.g., d-dimensional van der Corput-Halton or d-dimensional Kronecker sequences. We prove a necessary and sufficient condition for the higher-dimensional weighted q-adic sum-of-digits functions to be uniformly distributed modulo one in terms of a trigonometric product. As applications of our condition we prove some upper estimates of the extreme discrepancies of such sequences, and that the existence of distribution function g(x)= x implies the uniform distribution modulo one of the weighted q-adic sum-of-digits function sq,γ (n), n = 0, 1, 2,... We also prove the uniform distribution modulo one of related sequences h 1 sq, γ (n)+h 2 sq,γ (n +1), where h 1 and h 2 are integers such that h 1 + h 2 ≠ 0 and that the akin two-dimensional sequence sq,γ (n), sq,γ (n +1) cannot be uniformly distributed modulo one if q ≥ 3. The properties of the two-dimensional sequence sq,γ (n),s q,γ (n +1), n =0, 1, 2,..., will be instrumental in the proofs of the final section, where we show how the growth properties of the sequence of weights influence the distribution of values of the weighted sum-of-digits function which in turn imply a new property of the van der Corput sequence.

On uniform distribution of αβ-orbits

Let α,β∈(0,1) such that at least one of them is irrational. We take a random walk on the real line such that the choice of α and β has equal probability 1/2. We prove that almost surely the αβ-orbit is uniformly distributed module one, and the exponential sums along its orbit have the square root cancellation. We also show that the exceptional set in the probability space, which does not have the property of uniform distribution modulo one, is large in the terms of topology and Hausdorff dimension.

Обобщение определения среднего

В статье приводится решение задачи нахождения общего вида среднего в случае отсутствия симметричности по всем переменным. В 1930 году А. Н. Колмогоров дал общий вид среднего значения. Он сформулировал четыре аксиомы среднего: непрерывность и монотонность по каждой переменной, симметричность по каждой переменной, равенство среднего от одинаковых значений этому значению и возможность замены некоторой группы значений их собственным средним без изменения общего среднего. Все переменные в теореме Колмогорова равноправны, это предполагает, что среднее является сиимметрической функцией по всем переменным. В. Н. Чубариковым была поставлена задача обобщения результата А. Н. Колмогорова на случай отсутствия симметричности по всем аргументам. Теперь переменные разбиваются на группы, и среднее будет симметрично отдельно по каждой из групп переменных. Если такая группа единственна, то исследуемое среднего удовлетворяет аксиомам А. Н. Колмогорова, поэтому результат статьи является обощением теоремы Колмогорова. В статье найден общий вид функции среднего в этой задаче, отмечена связь с равномерным распределением по модулю единица.

Morphic words and equidistributed sequences

The problem we consider is the following: Given an infinite word w on an ordered alphabet, construct the sequence νw=(ν[n])n, equidistributed on [0,1] and such that ν[m]<ν[n] if and only if σm(w)<σn(w), where σ is the shift operation, erasing the first symbol of w. The sequence νw exists and is unique for every word with well-defined positive uniform frequencies of every factor, or, in dynamical terms, for every element of a uniquely ergodic subshift. In this paper we describe the construction of νw for the case when the subshift of w is generated by a morphism of a special kind; then we overcome some technical difficulties to extend the result to all binary morphisms. The sequence νw in this case is also constructed with a morphism.At last, we introduce a software tool which, given a binary morphism φ, computes the morphism on extended intervals and first elements of the equidistributed sequences associated with fixed points of φ.

A nonlocal transform to map and track quantum dynamics

The focus of this work is a correspondence between the Hilbert space operators on one hand, and doubly periodic generalized functions on the other. The linear map that implements it, referred to as the Q-transform, enables a direct application of the classical Harmonic analysis in a study of quantum systems. In particular, the Q-transform makes it possible to reinterpret the dynamic of a quantum observable as a (typically nonlocal) dynamic of a doubly periodic real function. From this point of view we carry out an analysis of an open quantum system whose dynamics are governed by an asymptotically harmonic Hamiltonian and compact type Lindblad operators. It is established that the initial value problem of the equivalent nonlocal but classical evolution is well posed in the appropriately chosen Sobolev spaces. The second set of results pertains to a generalization of the basic Q-transform and highlights a certain type of asymptotic redundancy. This phenomenon, referred to as the broadband redundancy, is a consequence of a well-known property of the zeros of the Riemann zeta function, namely, the uniform distribution modulo one of their ordinates. Its relevance to the analysis of quantum dynamics is only a special instance of its utility in harmonic analysis in general. It remains to be seen if the phenomenon is significant also in the physical sense, but it appears well-justified—in particular, by the results presented here—to pose such a question.

Discrete universality of the Riemann zeta-function and uniform distribution modulo 1

Discrete universality of the Riemann zeta-function and uniform distribution modulo 1

The discrepancy of Farey series

We study the uniform distribution modulo one of the Farey series in an arbitrary subinterval of the closed unit interval [0, 1], whose fractions have denominators streaming in a given arithmetic progression, and we establish upper and lower bounds for the discrepancy of the Farey series. The distribution is closely related to the growth of the Mertens function and follows from a concise formula.

Construction of Uniformly Distributed Linear Recurring Sequences Modulo Powers of 2

AbstractThe aim of the present paper is to provide the background to construct linear recurring sequences with uniform distribution modulo 2s. The theory is developed and an algorithm based on the achieved results is given. The constructed sequences may have arbitrary large period length depending only on the computational power of the used machines.

Bounded gaps between products of distinct primes

Let r ge 2 be an integer. We adapt the Maynard–Tao sieve to produce the asymptotically best-known bounded gaps between products of r distinct primes. Our result applies to positive-density subsets of the primes that satisfy certain equidistribution conditions. This improves on the work of Thorne and Sono.

Riemann’s zeta function and the broadband structure of pure harmonics

Let $a\in (0,1)$ and let $F_s(a)$ be the periodized zeta function that is defined as $F_s(a) = \sum n^{-s} \exp (2\pi i na)$ for $\Re s >1$, and extended to the complex plane via analytic continuation. Let $s_n = \sigma_n + it_n, \, t_n >0 $, denote the sequence of nontrivial zeros of the Riemann zeta function in the upper halfplane ordered according to nondecreasing ordinates. We demonstrate that, assuming the Riemann Hypothesis, the Cesaro means of the sequence $F_{s_n} (a)$ converge to the first harmonic $\exp (2\pi i a)$ in the sense of periodic distributions. This reveals a natural broadband structure of the pure tone. The proof involves Fujii's refinement of the classical Landau theorem related to the uniform distribution modulo one of the nontrivial zeros of $\zeta$.

On Weyl products and uniform distribution modulo one

In the present paper we study the asymptotic behavior of trigonometric products of the form prod _{k=1}^N 2 sin (pi x_k) for N rightarrow infty , where the numbers omega =(x_k)_{k=1}^N are evenly distributed in the unit interval [0, 1]. The main result are matching lower and upper bounds for such products in terms of the star-discrepancy of the underlying points omega , thereby improving earlier results obtained by Hlawka (Number theory and analysis (Papers in Honor of Edmund Landau, Plenum, New York), 97–118, 1969). Furthermore, we consider the special cases when the points omega are the initial segment of a Kronecker or van der Corput sequences The paper concludes with some probabilistic analogues.

\(\Delta\)-Convergence and Uniform Distribution in Lacunary Sense

In this paper, by considering usual partition of \([0, \infty)\) \(\Delta\)-convergence of non-negative real valued sequences is defined. It is shown that every convergent sequence is \(\Delta\)-convergence but the converse is not true, in general. Besides, some basic properties of \(\Delta\)-convergence as well as the second part of this paper by using any lacunary sequences as a partition of non-negative real numbers, lacunary uniform distribution is defined and some inclusion result between uniform distribution modulo 1 and lacunary uniform distribution has been given.

Some distribution results of the Oppenheim continued fractions

We study some distribution properties of the Oppenheim continued fraction expansions. A Gauss–Kuzmin–Levy type theorem is established. Based on this, a Frechet law concerning the partial maxima of the growth rate of the digit sequence is derived, which extends previous work of Galambos and Philipp on the regular continued fraction expansion. Besides, a uniform distribution modulo 1 result is obtained.

Uniform Distribution Modulo 1 and the Joint Universality of Dirichlet L -functions

In the paper, we prove a joint universality theorem on the approximation of a collection of analytic functions by a collection of shifts of Dirichlet L-functions L(s + iτ, \( \chi \) j ), where \( \tau \) takes values from the set {k α: k = 0, 1, 2, . . . } with 0 < \( \alpha \) < 1. The proof of this theorem uses the theory of uniform distribution modulo 1.