Powers of 3/2

The n^{th} value shown in the plot here is Mod[(3/2)^{n}, 1]. Measurements suggest that these values are uniformly distributed in the range 0 to 1, but despite a fair amount of mathematical work since the 1940s, there has been no substantial progress towards proving this.

In base 6, (3/2)^{n} is a cellular automaton with rule

{a_, b_, c_} -> 3 Mod[a + Quotient[b, 2], 2] + Quotient[3 Mod[b, 2] + Quotient[c, 2], 2]

(Note that this rule is invertible.) Looking at u (3/2)^{n} then corresponds to studying the cellular automaton with an initial condition given by the base 6 digits of u. It is then possible to find special values of u (an example is 0.166669170371...) which make the first digit in the fractional part of u (3/2)^{n} always nonzero, so that Mod[u (3/2)^{n}, 1] > 1/6. In general, it seems that Mod[u (3/2)^{n}, 1] can be kept as large as about 0.30 (e.g. with u =0.38906669065...) but no larger.