A sequence of real numbers is equidistributed on a given interval if the probability of finding in any subinterval is proportional to the subinterval length. In particular, if denotes the fractional part of by then a sequence is said to be equidistributed modulo 1 or uniformly distributed modulo 1 if , for all ,
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 be a sequence of real numbers. The following are equivalent:
- The sequence is equidistributed mod 1.
- For each nonzero integer , we have
- For each Riemann-integrable we have
2. Asymptotic equidistribution mod 1
Lemma 2.1. If is a bounded sequence of real numbers such that for every fixed , , then is equidistributed mod 1.
Example. Taking in Lemma 2.1, we have
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 Riemann integrable in it implies that the sequence approaches equidistribution mod 1 as . 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 is said to be asymptotically equidistributed modulo one if is increasing and the sequence of ratios
approach uniform distribution modulo one as .
Theorem 2.3. The sequence , where and are positive constants, is asymptotically equidistributed mod 1 and therefore by Weyl’s Criterion we have
Thus the sequence of natural numbers, the sequence of primes , the sequence of composite numbers and the sequence of the imaginary parts of the non-trivial zeroes of the Riemann zeta function are asymptotically equidistributed mod 1.
Trivially if equidistributed modulo 1 then 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 be any irrational number and be any real number. We have
- H. Wely: The sequence is equidistributed mod 1.
- I. Vinogradov: The sequence is equidistributed mod 1
- E. Hlawka: The sequence 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 is also asymptotically equidistributed modulo 1 and is any irrational number then the sequence 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 where is any sequence equidistributed modulo 1. Now from the set let us remove all elements that satisfy the condition to obtain a new set . Clearly no element of will have the digit 2 in the first place after the decimal point and therefore the set no longer equidistributed modulo 1 because no element of will lie in the interval . In general we can remove from the set all the elements which begins with a given sequence of digits after the decimal point. The remaining elements of will form a partially equidistributed sequence.We call the set to be partially equidistributed modulo 1 because as we shall see below that the set shows properties analogous to the set 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 digits after the decimal point, we can have different possibilities. From these different possibilities let us form a sequence partially equidistributed modulo 1 such that the first digits of after the decimal point can take different values from the set .
Theorem 3.1. If is partially equidistributed modulo 1 and is the set of allowed values of the first digits after the decimal point then
Theorem 3.1 is a generalization of (3) of Theorem 1. If then set will contain all possible combinations of digits. Hence will be equidistributed modulo 1 and RHS of Theorem 3.1 reduces to
To be contd.