Absolutely Abnormal Numbers

Absolutely Abnormal Numbers

Abnormal Numbers

A normal number is one whose decimal expansion (or expansion to some base other than 10) contains all possible finite configurations of digits roughly with their expected frequencies. More formally, let N(α; b, a, x) = #{1 ≤ n ≤ x : the nth digit in the base-b expansion of α is a} (1) denote the counting function of the occurrences of the digit a (0 ≤ a < b) in the b-ary expansion of the real n...

A polynomial-time algorithm for computing absolutely normal numbers

We give an algorithm to compute an absolutely normal number so that the first i digits in its binary expansion are obtained in time polynomial in i; in fact, just above quadratic. The algorithm uses combinatorial tools to control divergence from normality. Speed of computation is achieved at the sacrifice of speed of convergence to normality.

Computing Absolutely Normal Numbers in Nearly Linear Time

A real number x is absolutely normal if, for every base b ≥ 2, every two equally long strings of digits appear with equal asymptotic frequency in the base-b expansion of x. This paper presents an explicit algorithm that generates the binary expansion of an absolutely normal number x, with the nth bit of x appearing after npolylog(n) computation steps. This speed is achieved by simultaneously co...

On absolutely normal numbers and their discrepancy estimate

The algorithm computes the first n digits of the expansion of x in base 2 after performing triple-exponential in n mathematical operations. It is well known that for almost all real numbers x and for all integers b greater than or equal to 2, the sequence {bx}j≥0 is uniformly distributed in the unit interval, which means that its discrepancy tends to 0 as N goes to infinity. In [6], Gál and Gál...