Hillel Furstenberg and Grigoriĭ Margulis have been announced as the winners of the 2020 Abel Prize. You can read the official announcement here. There is a news item about the prize on the AMS website. Needless to say, they have made tremendous contributions to mathematics. In this post, I will point out a few things about Furstenberg and Margulis from MathSciNet.

**Note**: Terry Tao has a short post on his blog, which points to other, longer posts he has made mentioning Furstenberg or Margulis.

**Note (added 3/19/2020)**: Kenneth Chang has an informative article in the New York Times about the Abel Prize and this year’s prize winners.

First, playfully, we can see that the MR collaboration distance between them is 3:

So Furstenberg has a Margulis number of 3, and, symmetrically, Margulis has a Furstenberg number of 3. The path between them is not unique:

### Hillel Furstenberg

According to the Mathematics Genealogy Project, Furstenberg has 20 students and 171 descendants so far. He was, himself, a student of Salomon Bochner at Princeton. In MathSciNet, Furstenberg has 67 publications and 3,460 citations. He has 17 coauthors, the most frequent of which are Yitzhak Katznelson and Benjamin Weiss.

Furstenberg’s most cited work in MathSciNet is his book based on his Porter Lectures at Rice University in 1978, published as

**MR0603625** ** **

Furstenberg, H.

Recurrence in ergodic theory and combinatorial number theory.

M. B. Porter Lectures. *Princeton University Press, Princeton, N.J.,* 1981. xi+203 pp. ISBN: 0-691-08269-3

His most cited paper is

**MR0213508** ** **

Furstenberg, Harry

Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation.

*Math. Systems Theory* **1 **1967 1–49.

Furstenberg gave a series of CBMS Lectures at Kent State University in 2011, which were published as

**MR3235463**

Furstenberg, Hillel

Ergodic theory and fractal geometry.

CBMS Regional Conference Series in Mathematics, 120. *American Mathematical Society, Providence, RI,* 2014. x+69 pp. ISBN: 978-1-4704-1034-6

My favorite paper by Furstenberg is

**MR0068566**

Furstenberg, Harry

**On the infinitude of primes.**

*Amer. Math. Monthly* 62 (1955), 353.

There is no review in MathSciNet, but whatever we could have said would have been longer than the paper, which is a one paragraph proof of the infinitude of primes using topology!

### Grigoriĭ Margulis

According to the Mathematics Genealogy Project, Margulis has 19 students and 56 descendants so far. He was a student of Yakov Sinaĭ, another winner of the Abel Prize. In MathSciNet, Margulis has 113 publications and 3,740 citations. He has 48 coauthors, the most frequent of which has been Dmitry Kleinbock. Margulis is famous for various things, the first of which may be his proof of the Oppenheim conjecture:

**MR0882782**

Margulis, Gregori Aleksandrovitch(2-AOS-IT)

Formes quadratriques indéfinies et flots unipotents sur les espaces homogènes. (French summary) [Indefinite quadratic forms and unipotent flows on homogeneous spaces]

*C. R. Acad. Sci. Paris Sér. I Math.* 304 (1987), no. 10, 249–253.

Margulis’s book

**MR1090825 ****
**Margulis, G. A.(RS-AOS-IT)

**Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 17.**

Discrete subgroups of semisimple Lie groups.

Discrete subgroups of semisimple Lie groups.

*Springer-Verlag, Berlin,*1991. x+388 pp. ISBN: 3-540-12179-X

is a true classic. Besides this book, Margulis’s most cited works in MathSciNet are

**MR0939574** ** **

Margulis, G. A.

Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators. **(Russian)**

*Problemy Peredachi Informatsii* **24 **(1988), no. 1, 51–60; *translation in*

*Problems Inform. Transmission* **24 **(1988), no. 1, 39–46

and

**MR1652916** ** **

Kleinbock, D. Y.(1-IASP); Margulis, G. A.(1-YALE)

Flows on homogeneous spaces and Diophantine approximation on manifolds. (English summary)

*Ann. of Math. (2)* 148 (1998), no. 1, 339–360.

#### Some reviews of Hillel Furstenberg’s work.

**MR0603625** ** **

Furstenberg, H.

Recurrence in ergodic theory and combinatorial number theory.

M. B. Porter Lectures. *Princeton University Press, Princeton, N.J.,* 1981. xi+203 pp. ISBN: 0-691-08269-3

28D05 (10K10 10L10 54H20)

This very readable book discusses some recent applications, due principally to the author, of dynamical systems and ergodic theory to combinatorics and number theory. It is divided into three parts. In Part I, entitled “Recurrence and uniform recurrence in compact spaces”, the author gives an introduction to recurrence in topological dynamical systems, and then proves the multiple Birkhoff recurrence theorem: If $T_1,\cdots,T_l$ are commuting maps of a compact metric space to itself, then there exist a point $x$ of the space and a sequence $n_r$ of integers tending to infinity such that $\lim_{r\rightarrow\infty}T_i^{n_r}x=x$ for $1\leq i\leq l$. From this theorem a multidimensional version of van der Waerden’s theorem on arithmetic progressions is deduced, and applications to Diophantine inequalities are given.

Part II carries the title “Recurrence in measure preserving systems”. After a short introduction to the relevant part of measure-theoretic ergodic theory, this section is devoted to a proof of the multiple recurrence theorem: If $T_1,T_2,\cdots,T_l$ are commuting measure-preserving transformations of a finite measure space $(X,B,\mu)$ and if $A\in B$ with $\mu(A)>0$, then $$ \liminf_{N\rightarrow\infty}\frac 1{N}\sum_{n=1}^N\mu(T_1^{-n}A\cap T_2^{-n}A\cap\cdots\cap T_l^{-n}A)>0 $$ (and hence $\mu(T_1^{-n}A\cap\cdots\cap T_l^{-n}A)>0$ for some $n\geq 1$). From this result the author deduces a multidimensional version of Szemerédi’s theorem on the existence of arbitrarily long arithmetic progressions in sequences of integers with positive density.

Part III, called “Dynamics and large sets of integers”, investigates the connections between recurrence in topological dynamics and combinatorial results concerning finite partitions of the integers (e.g., Hindman’s theorem, Rado’s theorem). Here the notion of proximality plays a central role.

In reading this book, the reviewer found that the first part tickled his imagination and made him want to continue, the second part provided a good deal of work and tested his technical ability, while the last part led him to imagine the future possibilities for research. An excellent work!

Reviewed by Michael Keane

**MR0213508** ** **

Furstenberg, Harry

Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation.

*Math. Systems Theory* **1 **1967 1–49.

28.70 (10.00)

The approach to ergodic theory in this remarkable paper is complementary to the one developed, mainly by the Russian school, associated with numerical and group invariants. In fact, the relationship investigated here between two measure-preserving transformations (processes) and between two continuous maps (flows) is disjointness, an extreme form of non-isomorphism. The concept seems rich enough to warrant quite a few papers, and these papers will no doubt be largely stimulated by the present one. An interesting aspect of the paper, apart from the new results it contains, is the entirely novel demonstration of a number of established theorems. The paper is divided into four parts: (I) Disjoint processes; (II) Disjoint flows; (III) Properties of minimal sets; (IV) A problem in diophantine approximation. Two processes $X,Y$ are said to be disjoint $(X\perp Y)$ if whenever they are homomorphic images (factors) of the same process $Z$, then there is a homomorphism of $Z$ onto $X\times Y$ which, when composed with the projections of $X\times Y$ to $X,Y$, yields the given homomorphisms. (The commutativity in the diagram of this definition is essential, as a quick examination of a process $X$ which is isomorphic to $X\times X$ will reveal.) An equivalent definition insists that the inverse images of the two Borel fields be independent. The disjointness of two flows is defined similarly (but of course there is no analogous second definition). Two processes (flows) are co-prime if they have no non-trivial common factor. Disjointness implies co-primeness. Definitions are given of Bernoulli processes $\scr B$, Pinsker processes $\scr P$ (with completely positive entropy), deterministic processes $\scr D$ (with zero entropy) without reference to entropy. A particularly interesting class is the class $\scr W$ of Weyl processes, which, in view of the author’s structural theorem [Amer. J. Math. 85 (1963), 477–515; MR0157368], is a measure-theoretic analogue of the class of distal flows. {In this connection the reviewer is a little puzzled by the omission of the condition that $\scr W$ be closed under inverse limits, for it seems that such a definition would still yield the result “Mixing processes are disjoint from Weyl processes” and would provide yet another proof of L. M. Abramov’s result [Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962), 513–530; MR0143040; translated in Amer. Math. Soc. Transl. (2) 39 (1964), 37–56; see MR0160698] that processes with quasi-discrete spectrum have zero entropy.} Disjointness relations are established between the various classes, but M. S. Pinsker’s result [Dokl. Akad. Nauk SSSR 133 (1960), 1025–1026; translated as Soviet Math. Dokl. 1 (1960), 937–938; MR0152628] $\scr P\perp\scr D$ is not proved. Two processes with positive entropy are not disjoint, in fact are not co-prime. This is a consequence of Sinaĭ’s weak isomorphism theorem, but it is good to see a proof which does not depend upon such a deep result. Part I ends with a discussion of the relationship between disjointness and a problem of filtering.

In Part II analogues of weak mixing $\scr W$ and ergodicity $\scr E$ are defined for flows. Distal flows $\scr D$ are those such that $T^{m_n}x\rightarrow z$, $T^{m_n}y\rightarrow z$ imply $x=y$. Flows $T$ with a dense set of periodic points and such that $T^n\ (n\neq 0)$ is ergodic, are denoted by $\scr F$. The main results: If two flows are disjoint, one must be minimal $(\scr M)$; $\scr F\perp\scr M$; $\scr W\times\scr M\subset\scr E$; $\scr W\perp\scr D\cap\scr M$.

Part III is devoted to an analysis of the smallness of minimal sets for endomorphisms of compact abelian groups, with special emphasis on the circle group and the endomorphism $Tz=z^n\ (n\neq 0)$. In fact this endomorphism is an $\scr F$ flow and as such every minimal set is “restricted” and therefore cannot be a basis for the group. If $A$ is a $T$ invariant (closed) subset of the circle, then the topological entropy of $T$ restricted to $A$ is the Hausdorff dimension of $A$ multiplied by $\log n$. {A reference to P. Billingsley [e.g., Ergodic theory and information, Wiley, New York, 1965; MR0192027] and other authors would have been appropriate here.}

An example of a minimal set with positive topological entropy is given (cf. F. Hahn and Y. Katznelson [Trans. Amer. Math. Soc. 126 (1967), 335–360; MR0207959] for a sharper result). The main result of the final Part IV says that if $\Sigma$ is a non-lacunary (multiplicative) semi-group of integers and if $\alpha$ is irrational, then $\{n\alpha\ \text{mod}\,1\colon n\in\Sigma\}$ is dense in the unit interval.

Reviewed by W. Parry

**MR0498471**

Furstenberg, Harry

Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions.

*J. Analyse Math.* 31 (1977), 204–256.

10L10 (10K10 28A65)

L’article est consacré à la démonstration du théorème suivant d’apparence anodine: Si $(X,\scr B,\mu\,T)$ est un système dynamique et si $B\in\scr B$ est de $\text{mesure}>0$, pour tout entier $k$ il existe un entier $n$ tel que $\mu(B\cap T^nB\cap\cdots\cap T^{(k-1)n}B)\geq 0$. La démonstration en est relativement aisée lorsque la transformation $T$ est faiblement de type mélangé (weakly mixing). Mais il en va tout autrement dans le cas général. Et c’est l’occasion pour l’auteur de développer des techniques dont le champ d’application dépasse très largement la démonstration du théorème ci-dessus. Citons en particulier (i) la construction de la suite distale d’un système dynamique permettant de représenter un système dynamique comme un fibré dont la base est presque-périodique et les fibres faiblement de type mélangé. La base admet d’autre part une suite de composition (donnée par la série distale) dont les quotients sont des espaces homogènes de groupes compacts, (ii) l’étude de la décomposition ergodique des mesures invariantes sur un système dynamique produit d’une famille finie de systèmes dynamiques, à l’aide de la notion de mesure produit conditionnel sur un quotient: en gros on cherche une désintégration de la mesure donnée en mesures produits.

Ces quelques indications ne sauraient rendre compte de toute la richesse de cet article dont la lecture s’impose tout d’abord pour les résultats démontrés mais plus encore pour les moyens d’analyse mis en oeuvre.

Reviewed by François Aribaud

#### Some reviews of Grigoriĭ Margulis’s work.

**MR1090825** ** **

Margulis, G. A.(RS-AOS-IT)

Discrete subgroups of semisimple Lie groups.

Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 17. *Springer-Verlag, Berlin,* 1991. x+388 pp. ISBN: 3-540-12179-X

22E40 (20Hxx 22-02 22D40)

In 1972, when M. S. Raghunathan’s book Discrete subgroups of Lie groups appeared [Springer, New York, 1972; MR0507234], the theory of lattices (i.e. discrete subgroups of finite covolume) in nilpotent and solvable Lie groups was pretty well understood (most of the important results being due to A. I. Malʹtsev, G. D. Mostow, H.-C. Wang and L. Auslander). Thanks to the work of H. Minkowski, C. L. Siegel, A. Borel and Harish-Chandra, many important aspects of arithmetic subgroups of arbitrary groups were also well understood: Godement’s compactness criterion had been proved and a nice fundamental domain for such lattices had been constructed by Borel and Harish-Chandra and then an intrinsic one by Borel. However, the study of arbitrary lattices in semisimple groups had just begun with the rigidity results of A. Selberg and A. Weil. The next important development in the area was the proof by D. Kazhdan and the author of Siegel’s hypothesis which says that given a real semisimple Lie group $G$ and a fixed Haar measure on it, there is a constant $c>0$ such that with respect to the given Haar measure, the covolume of any lattice in $G$ is at least $c$; and of Selberg’s conjecture which asserts that a noncocompact lattice in a real linear semisimple Lie group contains a nontrivial unipotent element. Raghunathan’s book gave an elegant treatment of all the above results except the construction of fundamental domains for arithmetic subgroups in arbitrary semisimple groups. The latter was described in Borel’s Introduction aux groupes arithmétiques [Hermann, Paris, 1969; MR0244260].

Since the appearance of Raghunathan’s book, there have been several very profound developments in the theory of lattices in semisimple groups and most of these developments are due to the author of the book under review. Among these are the proof of super-rigidity and $S$-arithmeticity of irreducible lattice in any group of the form $G=\prod^n_{i=1}G_i$, where $G_i$ is the group of rational points of a semisimple algebraic group $G_i$ defined over a nondiscrete locally compact field $k_i$ and $\sum k_i-{\rm rank}\,G_i\geq 2$, and also the proof of finiteness of the index of any noncentral normal subgroup of a lattice in such a $G$. The author set himself the task of presenting a complete and reasonably self-contained account of these results in the book under review. He has admirably succeeded in his task. The proofs of results on super-rigidity, arithmeticity and the noncentral normal subgroups of lattices in real semisimple Lie groups are also given in a monograph by R. J. Zimmer [Ergodic theory and semisimple groups, Birkhäuser, Basel, 1984; MR0776417]. However, in the present book these results are proved in the most natural general setting, the only exception being the proof of arithmeticity of lattices, where the author assumes that in case $k_i$ is of positive characteristic, $G_i$ is “admissible”. T. N. Venkataramana, adapting the author’s proof to arbitrary characteristic, gave a proof of arithmeticity without any restrictions on the $G_i$. This proof became available a little too late to be included in the book.

The book includes a chapter on density and ergodicity theorems which contains proofs of various generalizations of Borel’s density theorem, the strong approximation property and Mautner’s lemma. There is also a chapter devoted to Kazhdan’s property (T) which gives a detailed account, with complete proofs, of results on groups with property (T). Two chapters of the book are devoted to the existence theorems for equivariant measurable maps, including a basic theorem in this direction due to H. Furstenberg. These existence theorems are vital for the proof of super-rigidity.

The book is carefully written and the reviewer did not spot any gaps or errors in the proofs.

Reviewed by Gopal Prasad

**MR1652916** ** **

Kleinbock, D. Y.(1-IASP); Margulis, G. A.(1-YALE)

Flows on homogeneous spaces and Diophantine approximation on manifolds. (English summary)

*Ann. of Math. (2)* 148 (1998), no. 1, 339–360.

11J83 (22E40)

FEATURED REVIEW.

This important paper settles conjectures of Baker and Sprindzhuk in the theory of Diophantine approximation on manifolds formulated in the 1970s. We adopt the following notations: for ${\mathbf x}\in\mathbf R^n$ let $\|\mathbf x\|=\max_{1\le i\le n} |x_i|, \Pi(\mathbf x)=\prod_{i=1}^n |x_i|$. One says that a vector $\mathbf x\in\mathbf R^n$ is very well approximated (VWA) if for some $\epsilon>0$ there are infinitely many $q\in\mathbf Z$ and $\mathbf p\in\mathbf Z^n$ such that $\|q\mathbf x+\mathbf p\|^n\cdot |q|\le|q|^{-\epsilon}$. Similarly, a vector $\mathbf x\in\mathbf R^n$ is very well multiplicatively approximated (VWMA) if for some $\epsilon>0$ there are infinitely many $q\in\mathbf Z$ and $\mathbf p\in\mathbf Z^n$ such that $\Pi(q\mathbf x+\mathbf p)\cdot |q|\le|q|^{-\epsilon}$.

Clearly, a vector is VWMA whenever it is VWA. One can also show that almost every $\mathbf x\in\mathbf R^n$ is not VWMA. A much more difficult problem arises if one restricts one’s attention to a proper submanifold $M\subset\mathbf R^n$. In the 1930s Mahler conjectured that almost all points on the curve $M_0=\{(t,t^2,\cdots,t^n), t\in\mathbf R\}\subset\mathbf R^n$ are not VWA. This was settled by V. G. Sprindzhuk [Dokl. Akad. Nauk SSSR 155 (1964), 54–56; MR0158868]. Later, Sprindzhuk in [Uspekhi Mat. Nauk 35 (1980), no. 4(214), 3–68, 248; MR0586190] conjectured the following. Let $f_1,\cdots,f_n$ be real-analytic functions in $\mathbf x\in U$, $U$ being a domain in $\mathbf R^d$, which together with $1$ are linearly independent over $\mathbf R$. Then almost all points of $M$ (in the sense of the natural measure on $M$) are not VWA.

The case $n=2$ of the conjecture was settled by W. M. Schmidt [Monatsh. Math. 68 (1964), 154–166; MR0171753], and recently V. V. Beresnevich and V. I. Bernik [Acta Arith. 75 (1996), no. 3, 219–233; MR1387861] proved it for $n=3$.

A stronger conjecture (also formulated by Sprindzhuk) states that almost all points of $M$ are not VWMA. It has not been proved even for the curve $M_0$ as above (this particular case is known as Baker’s conjecture) except for the case $n=2$ [V. G. Sprindzhuk, Metric theory of Diophantine approximations, Translated from the Russian and edited by Richard A. Silverman, Winston, Washington, D.C., 1979; MR0548467].

In the paper under review the authors prove a very general result which settles both of the conjectures of Sprindzhuk and Baker. Let $M=\{\mathbf f(\mathbf x)\colon\ \mathbf x\in U\}$, where $U$ is open in $\mathbf R^d$ and $\mathbf f=(f_1,\cdots, f_n)$ is a $C^m$ embedding of $U$ into $\mathbf R^n$. One says that a point $\mathbf y=\mathbf f(\mathbf x)$ of the manifold $M$ is nondegenerate if for some $l\le m$ the space $\mathbf R^n$ is spanned by partial derivatives of $\mathbf f$ at $\mathbf x$ of order up to $l$ (this is an infinitesimal version of not lying in any proper affine hyperplane). Assume now that almost every point of $M$ is nondegenerate. Then almost all points of $M$ are not VWMA.

The proof makes use of the ergodic theory of homogeneous actions on the space $\scr L_{n+1}={\rm SL}_{n+1}(\mathbf R)/{\rm SL}_{n+1}(\mathbf Z)$ (which is just the space of unimodular lattices in $\mathbf R^{n+1}$). More precisely, given $\mathbf y\in\mathbf R^n$ let $\Lambda_{\bf y}=\left(\begin{array}{ccc} 1&\mathbf y^{\top}\\ 0&I_n\end{array}\right)\mathbf Z^{n+1}$ be a unimodular lattice in $\mathbf R^{n+1}$. For $\mathbf t=(t_1,\cdots,t_n), t_i\ge 0$, define $t=\sum_{i=1}^n t_i\text{ and } g_\mathbf t={\rm diag}(e^t,e^{-t_1},\cdots,e^{-t_n})\in {\rm SL}_{n+1}(\mathbf R).$ Finally, one introduces a function $\delta$ on $\scr L_{n+1}$ by $\delta(\Lambda)=\inf_{\mathbf v\in\Lambda-\{0\}}\|v\|$.

It is easy to prove that for any $\mathbf y\in\mathbf R^{n}$ which is VWMA there exist $\gamma>0$ and infinitely many $\mathbf t\in\mathbf Z_+^n$ such that $\delta(g_\mathbf t\Lambda_\mathbf y)\le e^{-\gamma t}$. Thus, in view of Borel-Cantelli, it is enough to prove that for any nondegenerate point $\mathbf y_0=\mathbf f(\mathbf x_0)\in M$ there is a neighbourhood $B$ of $\mathbf x_0\in U$ such that given $\gamma>0$, one has $\sum_{\mathbf t\in\mathbf Z^n_+}|E_\mathbf t|<\infty$, where $E_\mathbf t=\{\mathbf x\in B\colon \delta(g_\mathbf t\Lambda_{\mathbf f(\mathbf x)})\le e^{-\gamma t}\}$ and $|\cdot|$ stands for the Lebesgue measure.

This is an immediate corollary of the following nontrivial estimate: (1) $|E_\mathbf t|\le Ce^{-\gamma t/{dl}}|B|$ for some ball $B$ centered at $\mathbf x_0$, some $C>0$ and all $\mathbf t\in\mathbf Z_+^n$ with large enough $t$ (here $l\le m$ is taken from the nondegeneracy condition for $\mathbf y_0$). The estimate is obtained by refining proofs of earlier important results on nondivergence of unipotent flows on the space of lattices.

Namely, let $\{u_x, x\in\mathbf R\}\subset {\rm SL}_k(\mathbf R)$ be a one-parameter subgroup all of whose entries are polynomials in $x$ (such a subgroup is called unipotent). A remarkable result of Margulis [in Lie groups and their representations (Proc. Summer School, Bolyai, János Math. Soc., Budapest, 1971), 365–370, Halsted, New York, 1975; MR0470140] states that no orbit of the action of $u_\mathbf R$ on $\scr L_{k}$ goes to infinity. By Mahler’s criterion, this is equivalent to saying that given any $\Lambda\in\scr L_{k}$, one has $\delta(u_x\Lambda)\not\to 0$ as $x\to\infty$. Later, S. G. Dani [Ergodic Theory Dynam. Systems 6 (1986), no. 2, 167–182; MR0857195] proved more: For any $c>0$ and any $\Lambda\in\scr L_{k}$ there exists $\epsilon>0$ such that given any unipotent subgroup $u_\mathbf R\subset {\rm SL}_k(\mathbf R)$, one has (2) $|\{x\in [0,T]\colon \delta(u_x\Lambda)<\epsilon\}|\le cT.$ Similar estimates were found by N. A. Shah [Duke Math. J. 75 (1994), no. 3, 711–732; MR1291701] for any polynomial map $\mathbf R^d\to {\rm SL}_k(\mathbf R)$ in place of the polynomial homomorphism $u\colon \mathbf R\to {\rm SL}_k(\mathbf R)$.

Here the authors establish a quantitative relation between $c$ and $\epsilon$ in (2), thus sharpening the result of Dani: it turns out that $c=c_1(k)\epsilon^{1/k^2}$. The proof uses the notion of $(C,\alpha)$-good functions introduced by A. Eskin, S. Mozes and Shah [Geom. Funct. Anal. 7 (1997), no. 1, 48–80; MR1437473], and the fact that polynomials of degree at most $k$ are $(C_k,1/k)$ good. In a similar way they obtain (1) using the fact that in a neighbourhood of a nondegenerate point, linear combinations of $1,f_1,\cdots,f_n$ are $(C,1/dl)$-good.

Finally, the authors note that the methods of their paper can help in solving related problems on Diophantine approximations in the $p$-adic case.

Reviewed by Alexander Starkov