Foundations of the Theory of Uniform Distribution

The book Uniform distribution of sequences by Kuipers and Niederreiter, long out of print, has recently been made available again by Dover books. I came across a copy at the Borders bookstore in San Francisco and decided to give it a try (the price, as they say, was right). It turned out to be full of interesting results, so I decided to take some notes on what seemed to me to be the parts most worthy of memory. Uniform distribution – or, as it is now more often called, “equidistribution” – of various arithmetic, (and geometric, spectral,. . .) objects is a hot topic nowadays, and was the subject of a recent SMS-NATO conference in Montréal. It seems that there is not so much in the way of foundational material available online, and most contemporary treatments of equidistribution expose the foundations with such extreme brevity that mistakes can arise. One notable exception is the 12 page article [?] based on Andrew Granville’s opening lecture at the Montréal conference. Their article has the merits of (i) true brevity (it can be digested in one sitting leaving a full but not aching stomach) and (ii) having been written by two of the leading authorities of the day. In contrast [KN] possesses (ii), but – instead of (i) – (i’): in its 330 pages it gives a wonderfully careful and complete (especially if one reads the fine printed “notes”) presentation of the foundations of the subject and some of the most important work up until about 1970. These notes, I hope, possess (i”): a treatment which is more systematic and complete than [?] but shorter than [KN] and presented in somewhat more modern language and style. Unfortunately I cannot claim (ii); caveat emptor.