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2010 | OriginalPaper | Buchkapitel
Distribution
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For any real number u let [u] denote the integer part of u, and (u) the fractional part. Then u = [u] + (u), where the first summand is an integer, and the second satisfies 0 ≤ (u) < 1. If u is irrational, the fractional parts (ku), k = 1, 2,… move about the interval (0, 1) in what seems at first to be a random way, but in fact a good deal can be said about their distribution. In this chapter we shall prove theorems about the distribution of this and other sequences in the interval. The definition of equidistribution and the results of this chapter are due to H. Weyl. The methods and results presented here were chosen because they involve trigonometric series.