M. Levin’s construction of absolutely normal numbers with very low discrepancy

Abstract

Among the currently known constructions of absolutely normal numbers, the one given by Mordechay Levin in 1979 achieves the lowest discrepancy bound. In this work we analyze this construction in terms of computability and computational complexity. We show that, under basic assumptions, it yields a computable real number. The construction does not give the digits of the fractional expansion explicitly, but it gives a sequence of increasing approximations whose limit is the announced absolutely normal number. The n-th approximation has an error less than 2 2 −n . To obtain the n-th approximation the construction requires, in the worst case, a number of mathematical operations that is double exponential in n. We consider variants on the construction that reduce the computational complexity at the expense of an increment in discrepancy.

On the construction of absolutely normal numbers

“…His construction does not produce directly the binary expansion of the defined number x. Instead it produces a computable sequence of real numbers that converge to x and the computation of the N -th term requires doubleexponential (in N ) many operations including trigonometric operations, see [2].…”

Computable absolutely Pisot normal numbers

“…This construction was made computable by Becher and Figueira [4] who gave a recursive formulation of Sierpinski's construction. Other algorithms for constructing absolutely normal numbers are due to Turing [36] (see also Becher,Figueira and Picchi [5]), Schmidt [32] (see also Scheerer [28]) and Levin [20] (see also Alvarez and Becher [1]).…”

Computable absolutely normal numbers and discrepancies

“…), where the speed of ω → ∞ and the implied constant depend on the base. Recently, Alvarez and Becher [1] analyzed Levin's work with respect to computability and discrepancy. They show that Levin's construction can yield a computable absolutely normal number α with discrepancy O( (log N ) 3 N 1/2 ).…”