Convergence Estimates for the Distribution of Trailing Digits

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Abstract
This paper analyzes the distribution of trailing digits (tail end digits) of positive real floating-point numbers represented in arbitrary base β and randomly chosen from a logarithmic distribution. The analysis shows that the n th digit for n ≥ 2 is actually approximately uniformly distributed. The approximation depends upon both n and the base β . It becomes better as n increases, and it is exact in the limit as n ⇒ ∞. A table of this distribution is presented for various β and n , 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.

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