Is the sequence $(3/2)^n \mod 1$ dense in the unit interval?
IsIn the sequenceother direction, $(3/2)^n \mod 1$ dense in the unit interval?Mahler's 3/2 problem:
Do all elements of this sequence with large enough index $n$ lie in the interval $(0,1/2)$?
It is known that $\beta^n$ is uniformly distributed modulo one for almost all $\beta>1$, but explicit examples of $\beta$ for which density holds are not known. This question seems to originate in work of Weyl and Koksma on uniform distribution.
Update:Update: Since posting this answer I've attempted to find some references with which to flesh it out, with only modest success. The earlier paper I have identified which deals with this question directly is T. Vijayaraghavan's 1940 article On the fractional parts of the powers of a number, in which it is shown that the sequence $(3/2)^n \mod 1$ has infinitely many limit points. Mahler conjectured in 1968 that the answer to his question is negative. Jeffrey Lagarias' 1985 survey on the Collatz problem, The 3x + 1 Problem and Its Generalizations, includes a one-page overview of the literature on the distribution of this sequence. Flatto, Lagarias and Pollington subsequently proved that the diameter of the set of accumulation points is at least 1/33; Mahler's question would be answered in the negative if this is improved to "at least 1/2".