Nielsen, Johan Sejr Brinch1; Simonsen, Jakob Grue2
1 Department of Physics, Technical University of Denmark2 University of Copenhagen
We investigate the distribution of digits of large prefixes of the expansion of irrational algebraic numbers to different bases. We compute 2.318 bits of the binary expansions (corresponding to 2.33.108 decimals) of the 39 least Pisot-Vijayaraghavan numbers, the 47 least known Salem numbers, the least 20 square roots of positive integers that are not perfect squares, and 15 randomly generated algebraic irrationals. We employ these to compute the generalized serial statistics (roughly, the variant of the χ2-statistic apt for distribution of sequences of characters) of the distributions of digit blocks for each number to bases 2, 3, 5, 7 and 10, as well as the maximum relative frequency deviation from perfect equidistribution. We use the two statistics to perform tests at significance level α = 0.05, respectively, maximum deviation threshold α = 0.05. Our results suggest that if Borel's conjecture-that all irrational algebraic numbers are normal-is true, then it may have an empirical base: The distribution of digits in algebraic numbers appears close to equidistribution for large prefixes of their expansion. Of the 121 algebraic numbers studied, all numbers passed the maximum relative frequency deviation test in all considered bases for digit block sizes 1, 2, 3, and 4; furthermore, 92 numbers passed all tests up to block size 4 in all bases considered.
Mathematics of Computation, 2013, Vol 82, Issue 283, p. 1837-1858