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Diophantine Approximations

Ivan Niven
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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In 1960, Ivan Niven was the Hedrick Lecturer at the summer meeting of the MAA. This book is the eventual result. Published by Wiley in 1963, it has now been reprinted by Dover. Niven was a great expositor, and many of his books are classics; here he gives us an introduction to the problem of approximating irrational numbers by rationals.

The central question is easily stated: given an irrational number x and an increasing real-valued function f, is it possible to find infinitely many rational numbers a/b such that

|xa/b| < 1/f(b)? 

The answer turns out, of course, to be "it depends." It can be done for every x if f(b) = b2, but not for f(b) = b2+ε for any ε > 0, and Niven gives an elegant proof of this. One can then try to determine the best constant K so that it can be done (for any x) with f(b) = Kb2. Or one can try to specialize to specific kinds of real numbers x. It turns out that whether x is algebraic or transcendental makes a huge difference.

Niven's account of the theory focuses on the general results (true for any x) rather than on the connections with transcendence theory. The proofs are elementary; in fact, Niven even avoids using continued fractions, which are normally the crucial tool. The elementary nature of his proofs allow him to generalize many of his results to approximations of complex numbers by numbers in Q(i), which he does in the last two chapters.

The middle chapter presents some neat results in a slightly different vein. (Niven points out that these results have been frequently rediscovered; I can certainly see that.) Take an irrational number α and consider the set of integers Nα consisting of the integer parts of all the integer multiples of α:

Nα = {[nα], n ∈  Z}.

It's easy to see that if α < 1 this is just all of Z, but for large α it is not at all clear which integers these are. Given m, are they equidistributed among the various congruence classes modulo m? For which pairs of real numbers (α, β) is it true that Nα and Nβ partition Z into two disjoint subsets? The answers are in the book.

These are all very nice results, and they are clearly explained and cleanly proved. What Niven gives no hint of is that they are also useful. Evem small hints about how these results fit into the rest of mathematics might have been helpful for those readers who find it a challenge to read a multi-page proof.

That worry aside, this is the sort of book one can give to an interested and motivated undergraduate. It will show them some beautiful mathematics that should lead them quickly into deeper waters.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College and the editor of MAA Reviews.

Notation and Conventions
1. The Approximation of Irrationals by Rationals
2. The Product of Linear Forms
3. The Multiples of an Irrational Number
4. The Approximation of Complex Numbers
5. The Product of Complex Linear Forms