Hardy and Littlewood proved that the sequence {frac(x^n)}, where frac(x) is the fractional part, is equidistributed for almost all real numbers x>1 (i.e., the exceptional set has Lebesgue measure zero). Exceptional numbers include the positive integers, the silver ratio 1 + sqrt(2), and the golden ratio ϕ. The plots above illustrate the distribution of frac(x^n) for x = e, π, ϕ, and 1 + sqrt(2). Candidate members of the measure one set are easy to find, but difficult to prove. However, Levin has explicitly constructed such an example.
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