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Metric number theory. (English) Zbl 1081.11057
London Mathematical Society Monographs. New Series 18. Oxford: Clarendon Press (ISBN 0-19-850083-1/hbk). xviii, 297 p. £ 75.00 (1998).

The book deals with properties of real numbers and with properties of $k$-tuples of real numbers which are not necessarily valid for all numbers (or $k$-tuples) but for “almost all” in some sense. The first result in this area was a now classical theorem of Borel, stating the following: Let $t\in \left[0,1\right)$, $q\in {ℤ}^{+}$, $q\ge 2$. We have the $q$-adic expansion of $t$: $t=\sum {c}_{k}\left(t\right){q}^{-k}$, where for each $k$, ${c}_{k}\left(t\right)$ is one of the numbers $0,1,\cdots ,q-1$. If we exclude the possibility ${c}_{k}\left(t\right)=q-1$ for all sufficiently large $k$, the representation is unique. For $j=0,1\cdots ,q-1$, denote ${N}_{n,j}\left(t\right)=#\left\{1\le k\le n$; ${c}_{k}\left(t\right)=j\right\}$. Then we have for almost all $t$ (that is, for all $t$, except possibly a set of Lebesgue-measure 0) ${N}_{n,j}\left(t\right)=\frac{m}{q}\left(1+o\left(1\right)\right)$ as $n\to \infty$. Since the ${c}_{k}\left(t\right)$’s are mutually independent random variables, the above result is a consequence of the strong law of large numbers. Since $q$ can be replaced by any power of $q$ and the $m$th power of $q$ corresponds to a configuration of $m$ digits, Borel’s theorem also states that for almost all $t$, each configuration of $m$ digits occur with the same asymptotic relative frequency $1/{q}^{m}$.

The first chapter deals with the above theorem and with the presentation of tools of probability theory needed for the sequel. The next chapter considers Diophantine approximation. After proving the classical Dirichlet theorem and mentioning Hurwitz’ improvement, the author discusses Khintchine’s theorem, according to which for monotonically decreasing $x\psi \left(x\right)$ the necessary and sufficient condition for the infinitely many times sovlability of $\left(*\right)$ $|\alpha -m/n|<\psi \left(n\right)/n$ for almost all $\alpha$ is the divergence of the series $\sum \psi \left(n\right)$. One can ask about the condition for the solvability of $\left(*\right)$, if the monotonicity of $x\psi \left(x\right)$ is relaxed. A still open conjecture of Duffin and Schaeffer is that for the solvability of $\left(*\right)$ the divergence of the series $\sum \psi \left(n\right)\phi \left(n\right)/n$ suffices. (Here $\phi \left(·\right)$ is the Euler function.) Relaxations of the original monotonicity restriction are proved by several authors. The proofs are based on probability theory (the 0 or 1 law, Borel-Cantelli lemma, estimates of certain subintervals of $\left[0,1\right)$).

Chatper 3 considers inhomogeneous Diophantine approximation in a slightly more general form, namely, the number of the solutions for almost all $\alpha$ of the inequality $\parallel \alpha {a}_{n}+{\beta }_{n}\parallel , where ${a}_{n}\in {ℤ}^{+}$, ${b}_{n}\in ℝ$, and $f\left(n\right)\ge 0$, $\sum f\left(n\right)=\infty$. Chapter 4 is devoted to the homogeneous problem having applications to the metrical theory of continued fractions and Chapter 5 to uniform distribution. The very interesting book concludes with the discussion of the Hausdorff dimensions of the occurring exceptional sets.

##### MSC:
 11K60 Diophantine approximation (probabilistic number theory) 11-02 Research monographs (number theory) 11J83 Metric theory of numbers 11Jxx Diophantine approximation 11J71 Distribution modulo one 11K55 Metric theory of other number-theoretic algorithms and expansions 11K50 Metric theory of continued fractions