Chapter 4: Systems Based on Numbers

Section 2: Elementary Arithmetic

General powers [of numbers]

It has been known in principle since the 1930s that Mod[hn, 1] is uniformly distributed in the range 0 to 1 for almost all values of h. However, no specific value of h for which this is true has ever been explicitly found. (Some attempts to construct such values were made in the 1970s.) Exceptions are known to include so-called Pisot numbers such as GoldenRatio, 2 + 1 and Root[#3 - # - 1 &, 1] (the numerically smallest of all Pisot numbers) for which Mod[hn, 1] becomes 0 or 1 for large n. Note that Mod[x hn, 1] effectively extracts successive digits of x in base h (see pages 149 and 919).

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From Stephen Wolfram: A New Kind of Science [citation]