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Asymptotic and partial equidistribution modulo one

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1. Introduction

A sequence of real numbers a_n is equidistributed on a given interval if the probability of finding a_n in any subinterval is proportional to the subinterval length. In particular, if \{a_n\} denotes the fractional part of a_n by then a sequence a_n is said to be equidistributed modulo 1 or uniformly distributed modulo 1 if , for all 0 \le b < c \le 1,

\displaystyle{\lim_{N \rightarrow \infty}\frac{\#\{r \le N : b < \{a_n\} \le c\}}{N} = c - b.}

Often problems related to equidistribution modulo 1 can be established using a powerful theorem called Weyl’s criterion which gives the necessary and sufficient conditions for a sequence to be equidistributed mod 1. Weyl’s criterion can be stated in the following equivalent forms.

Theorem 1. (Weyl’s Criterion) Let a_n be a sequence of real numbers. The following are equivalent:

  1. The sequence a_n is equidistributed mod 1.
  2. For each nonzero integer k, we have \displaystyle{\lim_{n \rightarrow \infty}\frac{1}{n}\sum_{r<n}e^{2\pi i k a_r } = 0}.
  3. For each Riemann-integrable f:[0,1] \rightarrow \mathbb{C} we have

\displaystyle{\lim_{n \rightarrow \infty}\frac{1}{n}\sum_{r \le n}f(\{a_r\}) = \int_{0}^{1} f(x)dx}.

2. Asymptotic equidistribution mod 1

Lemma 2.1. If u_n is a bounded sequence of real numbers such that for every fixed \epsilon, 0 < \epsilon \le 1\lim_{n \rightarrow \infty}u_{[n\epsilon]} = \epsilon then u_n is equidistributed mod 1.

Example. Taking u_r = r/n in Lemma 2.1, we have

\displaystyle{\lim_{n \rightarrow \infty}\frac{1}{n}\sum_{r \le n}f\Big(\frac{r}{n}\Big) = \int_{0}^{1} f(x)dx}.

Recall that the above example, also known as the rectangle formula, is a standard method in elementary calculus for evaluating the limit of a sum using definite integral. Since the rectangle formula holds for all functions f Riemann integrable in (0,1) it implies that the sequence r/n approaches equidistribution mod 1 as n \rightarrow \infty. This leads us to the concept of asymptotic equidistribution mod 1 which is defined as follows.

Definition 2.2. A sequence of positive real numbers s_n is said to be asymptotically equidistributed modulo one if s_n is increasing and the sequence of ratios

\displaystyle{\frac{s_1}{s_n},\frac{s_2}{s_n},...,\frac{s_n}{s_n}}

approach uniform distribution modulo one as n \rightarrow \infty.

Theorem 2.3. The sequence s_n = a n(\ln (b n))^{c}, where a, b and c are positive constants, is asymptotically equidistributed mod 1 and therefore by Weyl’s Criterion we have

\displaystyle{\lim_{n \rightarrow \infty}\frac{1}{n}\sum_{r \le n}f\Big(\frac{s_r}{s_n}\Big) = \int_{0}^{1} f(x)dx}.

Thus the sequence of natural numbers, the sequence of primes p_n, the sequence of composite numbers c_n and the sequence of the imaginary parts of the non-trivial zeroes of the Riemann zeta function \gamma _n are asymptotically equidistributed mod 1.

Example.

\displaystyle{\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{r \le n}\frac{p_n ^{2}}{p_n ^{2}+ p_r ^{2}}=\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{r \le n}\frac{\gamma _n ^{2}}{\gamma _n ^{2}+ \gamma _r ^{2}} = \frac{\pi}{4}.}

Trivially if a_n equidistributed modulo 1 then a_n is also asymptotically equidistributed modulo one. The converse of this however is highly non trivial and to appreciate this fact we shall look at three celebrated results of analytic number theory in the field of equidistribution mod 1.

Let \alpha be any irrational number and \beta be any real number. We have

  1. H. Wely: The sequence \alpha n is equidistributed mod 1.
  2. I. Vinogradov: The sequence \alpha p_n is equidistributed mod 1
  3. E. Hlawka: The sequence \beta \gamma_n is equidistributed mod 1.

This three results would follow as special cases if the following conjecture on asymptotic equidistribution were true.

Conjecture 2.4. If a_n is also asymptotically equidistributed modulo 1 and \alpha is any irrational number then the sequence \alpha a_n is equidistributed modulo 1.

A proof or disproof of this conjecture has eluded me so far.

3. Partial equidistribution modulo 1

Consider any the set S_0 = {a_1, a_2, ... , a_n} where a_n is any sequence equidistributed modulo 1. Now from the set S_0 let us remove all elements  that satisfy the condition 0.2 \le a_i < 0.3 to obtain a new set S_1. Clearly no element of S_1 will have the digit 2 in the first place after the decimal point and therefore the set S_1 no longer equidistributed modulo 1 because no element of S_0 will lie in the interval (0.2,0.3[. In general we can remove from the set S_0 all the elements which begins with a given sequence of digits after the decimal point. The remaining elements of S_1 will form a partially equidistributed sequence.We call the set S_1 to be partially equidistributed modulo 1 because as we shall see below that the set S_1 shows properties analogous to the set S_0 which is equidistributed modulo 1.

If numbers are represented in decimal nation, the first digit after the decimal point can have 10 different values, the first two digits after the decimal point can have 100 different values and in general, if we consider the first m digits after the decimal point, we can have 10^m different possibilities.  From these 10^m different possibilities let us form a sequence b_n partially equidistributed modulo 1 such that the first m digits of b_r after the decimal point can take k different values from the set D=\{d_1, d_2, ... , d_k\}.

Theorem 3.1. If b_r is partially equidistributed modulo 1 and D=\{d_1, d_2, ... , d_k\} is the set of allowed values of the first m digits after the decimal point then

\displaystyle{\lim_{n \rightarrow \infty}\frac{1}{n}\sum_{r=1}^{n}f(b_r) = \frac{10^m}{k} \sum_{d \in D}\int_{\frac{d}{10^m}}^{\frac{d+1}{10^m}}f(x)dx}.

Theorem 3.1 is a generalization of (3) of Theorem 1. If k=10^m then set D will contain all possible combinations of m digits. Hence b_r will be equidistributed modulo 1 and RHS of Theorem 3.1 reduces to

\displaystyle{\int_{0}^{1} f(x)dx}.

To be contd.

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