Convergence Estimates for the Distribution of Trailing Digits

This paper analyzes the distribution of trailing digits (tail end digits) of positive real floating-point numbers represented in arbitrary base <italic>β</italic> and randomly chosen from a logarithmic distribution. The analysis shows that the <italic>n</italic>th digit for <italic>n</italic> ≥ 2 is actually approximately uniformly distributed. The approximation depends upon both <italic>n</italic> and the base<italic>β</italic>. It becomes better as <italic>n</italic> increases, and it is exact in the limit as <italic>n</italic> ⇒ ∞. A table of this distribution is presented for various β and <italic>n</italic>, along with a table of the maximum digit by digit deviation Δ of the logarithmic distribution from the uniform distribution. Various asymptotic results for Δ are included. These results have application in resolving open questions of Henrici, of Kaneko and Liu, and of Tsao.