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Distribution modulo one and Diophantine approximation. (English) Zbl 1260.11001
Cambridge Tracts in Mathematics 193. Cambridge: Cambridge University Press (ISBN 978-0-521-11169-0/hbk; 978-1-139-53365-2/ebook). xvi, 300 p. £ 50.00; $80.00;$ 64.00 (2012).
The topic of this book is well described by the following three questions stated in the preface. Is there a transcendental real number $\alpha$ such that $||\alpha^n||$ tends to zero as $n$ tends to infinity? (Throughout $||y||$ stands for the distance from a real number $y$ to the nearest integer.) Is the sequence of fractional parts $\{(3/2)^n\}$, $n \geq 1$, dense in the unit interval? What can be said on the digital expansion of an irrational algebraic number? Of course, the first two of these questions are very far from being answered (the first is attributed to Hardy, 1919) and the research mainly concentrates on related questions, e.g., the distribution of $||\xi \alpha^n||$, $n \geq 1$, for algebraic $\alpha>1$ and real $\xi \ne 0$, Pisot and Vijayaraghavan numbers, Salem numbers, Furstenberg’s theorem, Mahler’s $Z$-numbers. The book begins with the Weyl criterion, some metrical results, discrepancy and distribution functions. Since this topic is described in detail in the book of [{\it L. Kuipers} and {\it H. Niederreiter}, Uniform distribution of sequences. Pure and Applied Mathematics. New York etc.: John Wiley & Sons (1974; Zbl 0281.10001)], the author gives only basic things and turns his attention to the sequences $||\xi \alpha^n||$, $n \geq 1$, and $\{\xi \alpha^n\}$, $n \geq 1$, where $\alpha>1$ and $\xi \ne 0$. These sequences are uniformly distributed for almost all $\xi$ (when $\alpha>1$ is fixed) and for almost all $\alpha>1$ (when $\xi \ne 0$ is fixed), but for most individual pairs $(\xi,\alpha)$ the problem is completely open. In the second chapter “On the fractional parts of powers of real numbers” the author gives some early results of Thue, Hardy, Pisot and Vijayaraghavan and describes the pairs $(\xi, \alpha)$ for which the fractional parts $\{\xi \alpha^n\}$, $n \geq 1$, avoid an interval of positive length in $[0,1]$. Among results that are presented with proofs Corollary 2.17 asserts that for each real number $\varepsilon$ satisfying $0<\varepsilon<1/20$ and each real number $\eta$ there exists a positive real number $\xi$ such that $\inf_{n \geq 1} ||\xi (1+\varepsilon)^n+\eta||$ exceeds $3 \cdot 10^{-3} \varepsilon |\log \varepsilon |$. It is also shown that the powers of some very special transcendental numbers $\alpha^n$ can approximate arbitrary sequence of real numbers $r_n$, $n \geq 1$ with any fixed precision, namely, $||\alpha^n -r_n||<\varepsilon$ for each $n \geq 1$. The results of this type can be proved by the Cantor-type construction and by an explicit construction of the reviewer. In this chapter the author also gives the proof of Furstenberg’s theorem on the density of $\{\xi r^m s^n\}$, $m,n \geq 0$, where $\xi$ is an irrational number and $r,s$ are multiplicatively independent integers, and describes some results towards so-called mixed Littlewood conjecture asserting that for every real number $\xi$ and every prime number $p$, we have $\inf_{q \geq 1} q ||\xi q|| |q|_p=0$. In the third chapter “On the fractional parts of powers of algebraic numbers” the author gives the proof of the reviewers estimate between the largest and the smallest limit points of the sequence $\{\xi \alpha^n+\eta\}$, $n \geq 1$, when $\alpha>1$ is an algebraic number [Bull. Lond. Math. Soc. 38, No. 1, 70--80 (2006; Zbl 1164.11025)]. Such estimate from below is possible when $\alpha$ is not a Pisot or a Salem number. (If it is then there is an extra condition that $\xi$ lies outside the field generated by $\alpha$.) Many other related results of the author, Pollington and the reviewer are also given with complete proofs. For instance, Theorem 3.12 gives sharp bounds on the largest and the smallest limit points of the sequence $||\xi b^n||$, $n \geq 1$, where $b \geq 2$ is an integer and $\xi$ is an irrational real number. In Chapter 4, entitled “Normal numbers”, the author turns into the problems of normality of real numbers. Recall that a real number $\xi$ is called normal in base $b$ if for each $k \geq 1$ every block of $k$ digits from $\{0,1,\dots,b-1\}$ occurs in the $b$-ary expansion of $\xi$ with the same frequency $b^{-k}$. Although, by an old result of Borel, almost all real numbers are normal to all integer bases, for concrete numbers like $e, \pi, \log 2$, the problem of normality is intractable. One can easily see that the problem of normality is related to uniform distribution, since the number $\xi$ is normal to base $b \geq 2$ if and only if the sequence $\xi b^n$, $n \geq 1$, is uniformly distributed modulo one (see Theorem 4.14 which is an observation of Wall). In this section the author also proves Mahler’s result that the Champernowne number $0.1234567891011121314\dots$ and some related numbers are transcendental and not Liouville numbers. In Chapter 5 the author proves that if $b, c \geq 2$ are coprime integers then the Stoneham number $\sum_{j \geq 1} c^{-j} b^{-c^j}$ and the Korobov number $\sum_{j \geq 1} c^{-d^j} b^{-c^{d^j}}$, where $d \geq 2$, are normal to base $b$. On the other hand, there are explicit constructions of numbers that are normal to no base. For instance, Theorem 5.8 (which is a result of {\it G. Martin} [Am. Math. Mon. 108, No. 8, 746--754 (2001; Zbl 1036.11035)]) asserts that the number $\prod_{j=2}^{\infty} (1-1/d_j)$, where $d_2=4$ and $d_j=j^{d_{j-1}/(j-1)}$ for $j \geq 3$, is a Liouville number which is simply normal to no base. At the end of this chapter the author discusses a hypothesis of Bailey and Crandall which (although appears intractable) would imply that each of the constants $\pi, \log 2, \zeta(3)$ is normal to base $2$. In Chapter 6 some problems of normality to different bases are investigated. In particular, the author proves the result of Schmidt asserting that if $r,s \geq 2$ are multiplicatively independent integers then the set of real numbers which are normal to base $r$ but not simply normal to base $s$ is uncountable. (The proof is given in the particular case $s=3$.) Some results of normality to non-integer bases are also discussed in this chapter. In Chapter 7 the author introduces and studies exponents of Diophantine approximation and gives some Diophantine results on the middle third Cantor set. Chapter 8 mainly deals with the digital expansions of algebraic irrational numbers in base $b$. In particular, the author gives a proof of his result with {\it B. Adamczewski} [Ann. Math. (2) 165, No. 2, 547--565 (2007; Zbl 1195.11094)] which asserts that $p(n,\xi,b)/n \to \infty$ as $n \to \infty$, where $p(n,\xi,b)$ stands for the number of distinct blocks of length $n$ in the $b$-ary expansion of the algebraic irrational number $\xi$. The expected value of $p(n,\xi,b)$ is $b^n$. A result of {\it D. Bailey} et al. [J. Théor. Nombres Bordx. 16, No. 3, 487--518 (2004; Zbl 1076.11045)] on the number of non-zero digits in the $b$-ary expansion of such $\xi$ is also reproduced. In Chapter 9 the author discusses similar questions for “complexity” of continued fraction expansions of algebraic irrational numbers and surveys various results on the so-called $\beta$-expansions (which is now an active area of research). Chapter 10 contains a list of 61 open questions which are collected from various papers. At the end of the book there are six appendices and more than forty pages of references. The reader may learn a lot from this book about various techniques used in this subject over many years (e.g., classical and metrical Diophantine approximation, combinatorics on words) and, in addition, find the proofs of some very recent results.

##### MSC:
 11-02 Research monographs (number theory) 11J71 Distribution modulo one 11K06 General theory of distribution modulo 1