Power Fractional Parts

Power fractional parts are values , where is the fractional part. Hardy and Littlewood (1914) proved that the sequence is equidistributed for almost all real numbers (i.e., the exceptional set has Lebesgue measure zero). Exceptional numbers include the positive integers, the silver ratio (Finch 2003), and the golden ratio . The plots above illustrate the distribution of for , , , and . Candidate members of the full-measure set are easy to find, but difficult to prove. However, Levin has explicitly constructed such an example (Drmota and Tichy 1997).

The properties of , the simplest such sequence for a rational number , have been extensively studied (Finch 2003). The first few terms are 0, 1/2, 1/4, 3/8, 1/16, 19/32, 25/64, 11/128, 161/256, 227/512, ... (OEIS A002380 and A000079; Pillai 1936, Lehmer 1941), plotted above (Wolfram 2002, pp. 121-122). For example, has infinitely many accumulation points in both and (Pisot 1938, Vijayaraghavan 1941). Furthermore, Flatto et al. (1995) proved that any subinterval of containing all but at most finitely many accumulation points of must have length at least 1/3. Surprisingly, the sequence is also connected with the Collatz conjecture and with Waring's problem.

Numbers of the form , where is the fractional part, appear in Waring's problem. In particular, Waring's problem can be solved completely if the inequality

holds for every integer . No counterexample to this inequality in that range is known, and it is even believed that it can be extended to

for (Bennett 1993, 1994, Finch 2003). These inequalities remain unproved over their entire stated ranges. The weaker two-sided inequality with 0.5769 in place of 3/4 has been proved for all sufficiently large (Dubitskas 1990; Habsieger 2003, p. 299).