RESEARCH STATEMENT DAVID SIMMONS

C ONTENTS 1.

Introduction

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2.

The potential game

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3.

Dimension gaps in sponges

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4.

Hausdorff dimension of uniformly badly approximable matrices

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5.

Well approximable matrices on fractals

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References

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1. I NTRODUCTION My main areas of research are Diophantine approximation, fractal geometry, dynamical systems, and the interactions between them. My research has been published in leading journals such as Advances in Mathematics,

Inventiones mathematicae, Journal de Math´ematiques Pures et Appliqu´ees, Mathematical Proceedings of the Cambridge Philosophical Society, Proceedings of the London Mathematical Society, and Selecta Mathematica. I also have longer work published in Mathematical Surveys and Monographs as well as in the Memoirs of the American Mathematical Society. In what follows I will begin with a brief overview of the fields mentioned above aimed at the non-specialist reader, and move on in separate sections to discuss four particularly significant discoveries of my research. I will only mention a small sample of threads that I intend to pursue in my future research. For a list of abstracts of all of my papers, as well as works in progress, please see “List of Publications, Preprints, and Projects in Progress” at https://sites.google.com/site/davidsimmonsmath/research/Project_List_2017.pdf. In its simplest case, Diophantine approximation is the study of the approximation of irrational numbers by rational numbers. Specifically, given an irrational number x, one asks how well x can be approximated by rational numbers p/q. The “quality” of an approximation p/q is measured in two ways: the “error” |x − p/q|, and the “height” q. There is a tradeoff between accuracy and height: one can find a sequence of rational numbers p/q whose errors converge to zero, but their heights must tend to infinity. Diophantine approximation is about quantifying this tradeoff, and describing how it varies over irrational numbers x (namely, some irrationals can be approximated better than others). Once one has this general picture of approximations, error, height, and tradeoff, one can begin to modify the picture, e.g. by extending it to higher dimensions or even to infinite dimensions [37], by considering other fields such as C or the field of p-adic numbers, or by using a nonstandard height function [36]. 1

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Fractal geometry is the study, within geometric measure theory, of sets and measures too complicated to study via classical methods. A key concept is the notion of fractal dimensions, which are ways of assigning a “dimension” to a set or measure which is not necessarily an integer. The word “fractal” was coined by Mandelbrot, from the Latin fractus meaning broken, who went on to give a “tentative definition” of a fractal as a set whose Hausdorff dimension is strictly greater than its topological dimension. Fractal dimensions provide an intuitive way of understanding the “size” of a set in the case where the set has Lebesgue measure zero. An important tool in fractal geometry is the theory of Schmidt games, which are two-player infinite games of perfect information such that the outcome of the game provides information about some fractal set. In many cases Schmidt games can be used to compute fractal dimensions of a set, and they also provide information about the size of the intersections of various sets. In various papers I have defined several new Schmidt games with applications to Diophantine approximation, see [20, 29, 33, 40]. Dynamical systems is a broad subject that includes the theories of iterated function systems and their generalizations, Kleinian groups and their generalizations, iteration of rational functions, and flows on homogeneous spaces. In many cases the dynamical system has an associated object called a “limit set” which we can then analyze using the tools of fractal geometry. Much of my work concerns conformal dynamical systems, in which the relevant dynamical maps are infinitesimally similarities, see [8, 12, 16, 17, 19, 22, 23, 24, 34, 35, 38, 39, 40, 62, 64]. However, I also study affine iterated function systems, in particular the fractal dimensions of self-affine sponges, see [18, 21]. Both Diophantine approximation and dynamical systems are connected to fractal geometry because they provide sets and measures which can be studied via fractal geometry. In particular, Schmidt games were originally developed as a solution to a problem in Diophantine approximation, namely showing that the intersection of certain images of the set of badly approximable numbers is nonempty. There are several different points of contact between the theories of Diophantine approximation and of dynamical systems. A simple yet profound question which I address in various papers [12, 16, 17, 18, 19, 31, 38, 40, 64] is: given sets and measures constructed from a dynamical system, what can we say about their Diophantine properties? In this framework, the Diophantine setup is viewed as being essentially independent of the dynamical system: the approximation set is the usual Qd , and the height function is simply the usual height function on Qd . This framework can be contrasted with a second way of connecting Diophantine approximation and dynamical systems, in which the dynamics are themselves used to create a Diophantine setup. One way to do this is to let the “approximating set” be the orbit of a distinguished point under a Kleinian group with an appropriate height function, see [8, 40, 62]. This point of view turns out to yield applications of Diophantine approximation to questions in fractal geometry which initially are unrelated to Diophantine approximation, see [24, 62]. Alternatively, one can let the “approximating set” be the set of rationals contained in the limit set of a dynamical system, with a new height function appropriate to the dynamics, see [8, 30, 32, 34]. A third connection between dynamics and Diophantine approximation comes from the so-called “correspondence principle”, introduced by Dani [15, Theorem 2.20], refined by Kleinbock–Margulis [47, Theorem 8.5], and extended to a new setting in my paper [28, §7]. The correspondence principle relates the Diophantine approximation properties of a point in Rn or on a quadratic variety to the dynamics of a corresponding orbit of a certain one-parameter flow on a homogeneous space. It can thus be used to deduce Diophantine theorems from dynamical ones; we use this strategy in [16, 20, 28, 63, 64].

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2. T HE POTENTIAL GAME A fundamental notion in fractal geometry is Hausdorff dimension, see e.g. [26]. One tool for computing Hausdorff dimension is the notion of a winning set, which we now define. Let X be a complete metric space. For any 0 < α, β < 1, Schmidt’s (α, β)-game is an infinite game played by two players, Alice and Bob, who take turns choosing balls in X, with Bob moving first. The players must choose their moves so as to satisfy the relations B1 ⊇ A1 ⊇ B2 ⊇ · · · and ρ(Ak ) = αρ(Bk ) and ρ(Bk+1 ) = βρ(Ak ) for k ∈ N, where Bk and Ak denote Bob’s and Alice’s kth moves, respectively, and where ρ(B) denotes the radius of a ball B. Since the sets B1 , B2 , . . . form a nested sequence of nonempty closed sets whose diameters tend to zero, it T T follows that the intersection k Bk is a singleton, say k Bk = {x∞ }. The point x∞ is called the outcome of the game. To determine the winner, we fix a set S ⊆ X (called the target set) and declare that Alice wins if x∞ ∈ S, and Bob wins if x∞ ∈ / S. The target set set S is said to be (α, β)-winning if Alice has a strategy guaranteeing her victory, regardless of the way Bob chooses to play. It is said to be α-winning if it is (α, β)-winning for every 0 < β < 1, and winning if it is α-winning for some 0 < α < 1. If the space X is sufficiently nice (e.g. Ahlfors regular, meaning that there is a measure µ such that µ(B(x, r))  rδ for all x ∈ X and 0 < r ≤ 1), then every winning set has full Hausdorff dimension [27]. Moreover, for each α > 0 the class of α-winning sets is closed under countable intersections, and the class of winning sets is closed under bi-Lipschitz maps [60]. These facts can be used to prove that various sets have full Hausdorff dimension, including the set of badly approximable matrices [61]. The basic scheme appearing in the definition of a winning set has seen many variants. These include the notions of absolute winning and strong winning sets introduced by McMullen [54], modified winning sets by Kleinbock and Weiss [49], and hyperplane absolute winning sets by Broderick, Fishman, Kleinbock, Reich, and Weiss [10]. My own contributions include the hyperplane percentage game [11], the algebraic set game [29], the Banach–Mazur– Schmidt game [33], the Hausdorff game [20], and finally the potential game [40, Appendix C]. Each notion of winning sets is based on a different game, which nevertheless shares the common feature that the gameplay is described as a sequence of geometric objects, whose point of intersection is called the outcome of the game and is compared to a fixed target set. Different notions of winning have different incompressibility properties; for example, absolute winning and strong winning sets are incompressible under quasisymmetric maps (see [54]), while hyperplane winning sets are only incompressible under C 1 maps (see [10]). All of them have applications to Diophantine approximation, and there are various examples of Diophantine sets winning for some games but not for others. I originally introduced the potential game in joint work with Lior Fishman and Mariusz Urba´ nski [40] in order to deal with a specific problem in Diophantine approximation, namely the problem of proving the large Hausdorff dimension of the set of badly approximable points of a strongly discrete group acting on a Gromov hyperbolic metric space. However, since then various researchers have used the game to solve a wide variety of Diophantine problems, see [1, 2, 3, 9, 12, 43, 55, 66, 67]. The game was also considered important enough to be included in the topics list for the Oberwolfach Arbeitsgemeinschaft in Diophantine approximation, fractal geometry, and dynamics in 2016.

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F IGURE 1. Three consecutive turns of the potential game. On each turn Bob chooses a ball (colored white) and then Alice “deletes” some number of proportionally smaller balls (colored red). Bob does not have to avoid the deleted balls immediately, but must do so eventually. To define the potential game (cf. [40, Definition C.4], and see Figure 1 for an illustration), let X be a complete metric space, let H be a collection of closed subsets of X, and fix β, c > 0. As in Schmidt’s game, Bob will choose a descending sequence of balls B1 ⊇ B2 ⊇ B3 ⊇ · · · satisfying ρ(Bk+1 ) ≥ βρ(Bk ). However, instead of choosing her own sequence of balls, Alice will react to Bob’s kth move Bk by choosing a countable collection Ak consisting of sets of the form N (L, ε) = {x ∈ X : d(x, L) ≤ ε} and satisfying
c εc ≤ βρ(Bk ) .

X N (L,ε)∈Ak

Alice’s choices have no immediate effect on the game, but they affect which player is declared the winner. Namely, if S ⊆ X is the target set, then Alice wins if x∞ ∈ S ∪

∞ [

[

N (L, ε)

k=1 N (L,ε)∈Ak

and Bob wins otherwise. It may happen that the radii ρ(Bk ) do not tend to 0 in which case the outcome x∞ is ill-defined; in this case we declare Alice the winner. The game described above is called the (β, c, H)-potential game, and the target set S is said to be (β, c, H)-potential winning if Alice has a strategy guaranteeing her victory. A set is called (c, H)-potential winning if it is (β, c, H)-potential winning for all β > 0.

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The name “potential game” comes from the fact that the proofs of many facts about the potential game involve a potential function, which is a function that measures how many obstacles are in a given region, weighted according to their size raised to the power of c. This function is useful because it can help Bob decide where to play: avoid the regions with the most obstacles. This general strategy can be used to prove many nice properties of potential winning sets. In particular, if a set is (c, H)-potential winning for every c > 0 then it is H-absolute winning [40, Theorem C.8] and if X is Ahlfors regular of dimension δ, then for every 0 < c < δ, every (c, P)potential winning set has full Hausdorff dimension in X, where P = {{x} : x ∈ X} (e.g. this follows from [12, Theorem 5.5]). The success of the potential game is that it allows one to deal with situations in Diophantine approximation where there are many small objects to be avoided. Previous games such as the absolute game and the hyperplane game dealt only with the issue of avoiding a single object, or of avoiding a collection of objects with a common geometrical property (i.e. all lying on a hyperplane). However, in many situations the objects which one wants to avoid do not have such a convenient geometric grouping, which is where the potential game becomes relevant. There are two main insights that explain the success of the potential game. The first is that when dealing with small objects, it is not necessary to avoid them immediately, as long as one avoids them eventually. The second is that one should try to avoid as many small objects as possible at any given time, with the word “many” incorporating an appropriate weighting on the set of objects. Once these insights are in place, a computation needs to be performed to check that it is possible to avoid the objects fast enough if there are not too many of them. Questions for future research. Question 2.1. Can the potential game be used to prove that the set Bad(ß) is hyperplane winning for every Pd vector ß = (s1 , . . . , sd ) ∈ [0, 1]d such that 1 si = 1? See [43] for a partial result with an additional assumption on the vector ß.

3. D IMENSION GAPS IN SPONGES An iterated function system on a compact set X ⊆ Rd is a collection of uniformly contracting similarities φa : X → X. If Φ = (φa )a∈E is an iterated function system, then the coding map π : E N → X is defined by the formula π(ω) = lim φω1 ◦ · · · ◦ φωn (x0 ), n→∞

where x0 ∈ X is an arbitrary point. The limit always exists due to the contraction assumption. Finally, the limit set of Φ is the image of the coding map: ΛΦ = π(E N ). The difficulty of computing the Hausdorff dimension of the limit set depends on what hypotheses one makes regarding the iterated function system Φ. If Φ is assumed to be a conformal iterated function system satisfying the open set condition, then the Hausdorff dimension of ΛΦ can be computed via Bowen’s formula, see [53]. However, if the condition of conformality is dropped, computing the Hausdorff dimension becomes much more difficult. In joint work with Tushar Das [21], I introduced a formula for computing the Hausdorff dimension of a certain class of limit sets of non-conformal iterated function systems known as self-affine sponges, see Figure 2. This formula was a surprise from the point of view of the existing literature, because it used the notion of a measure associated with an exponentially periodic function, whereas previous analogous formulas did not need such a notion. An exponentially periodic function is a continuous map b 7→ rb : (0, ∞) → P(E) such that there exists λ > 1 such that rλb = rb for all b > 0, where P(E) denotes the space of probability measures on the alphabet E.

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F IGURE 2. The first and second levels in the construction of a self-affine sponge. At each level, each box is replaced by four smaller boxes. Images by Douglas Howroyd.

The measure associated to the exponentially periodic function r is the measure Y µr = rn ∈ P(E N ). n∈N

Intuitively, the measure associated to an exponentially periodic function is a measure which behaves radically differently at different length scales, in contrast with more commonly considered measures (such as shift-invariant measures) which behave similarly at different length scales. We proved that the Hausdorff dimension of a selfaffine sponge ΛΦ is equal to the supremum of the Hausdorff dimensions of the measures associated to exponentially periodic functions (or more precisely, the images of these measures under the coding map) [21, Theorem 2.11]. We also gave a formula for the Hausdorff dimension of the measure associated to an exponentially periodic function [21, Theorem 2.15]. We were able to leverage the unexpected appearance of exponentially periodic functions in our formula to show that it implies the existence of self-affine sponges whose Hausdorff dimension is strictly greater than the “expected” dimension. One way of defining the “expected” dimension is the notion of dynamical dimension. The dynamical dimension of a self-affine sponge is the supremum of the Hausdorff dimensions of the measures on the sponge invariant under the shift map on E N . We proved that the Hausdorff dimension of a self-affine sponge in Rd can be strictly greater than its dynamical dimension whenever d ≥ 3 [21, Theorem 2.8]. This “dimension gap” result incidentally answered a problem open since the early 1990s, stated by Schmeling and Weiss to be “one of the major open problems in the dimension theory of dynamical systems” [59, p.440] (see also [52, 46, 41, 42, 56, 57, 14, 58, 4]): does every expanding repeller have an ergodic invariant measure of full Hausdorff dimension? The connection between this problem and our theorem is that a self-affine sponge is an expanding repeller, and if the dynamical dimension of a sponge is strictly less than its Hausdorff dimension, then it cannot have any invariant measures of full dimension. Thus not every expanding repeller has an ergodic invariant measure of full Hausdorff dimension. This means that there are expanding repellers whose limit sets do not have any “natural” measures, if one takes the view that a natural measure is an ergodic invariant measure of full Hausdorff dimension. Questions for future research.

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Question 3.1. What is the asymptotic behavior of the function def

f (d) =

sup ΛΦ


dimH (ΛΦ ) − dimD (ΛΦ ) ,

⊆[0,1]d

where the supremum is taken over all self-affine sponges Λ? Here dimD denotes the dynamical dimension. Our theorem implies that f (d) > 0 for all d ≥ 3. Question 3.2. Is the function Φ 7→ dimH (ΛΦ ) real-analytic, or at least piecewise real-analytic?

4. H AUSDORFF DIMENSION OF UNIFORMLY BADLY APPROXIMABLE MATRICES An m × n matrix A is said to be badly approximable if there exists a constant κ > 0 such that the inequality (4.1)

n kAq − pkm µ kqkν ≤ κ

has only finitely many integer solutions (p, q) ∈ Zm × (Zn \ {0}). Here k · kµ and k · kν denote arbitrary but fixed norms on Rm and Rn , respectively. Schmidt proved that the Hausdorff dimension of the set of badly approximable matrices is equal to mn [61], generalizing a result of Jarn´ık [45]. There have been many attempts to prove a quantitative version of the Jarn´ık–Schmidt theorem by estimating the Hausdorff dimension of the set B(κ) consisting of matrices A such that (4.1) has only finitely many integer solutions, see [51, 44, 65, 13]. Currently the most accurate estimate is my result proven using Cantor series expansions and the Dani correspondence principle, which states that dimH (B(κ)) = mn − θµ,ν κ + o(κ) where θµ,ν > 0 is an explicit constant depending on the norms k · kµ and k · kν (see [63]). This is the first result that has the correct order-of-magnitude estimate for the codimension mn − dimH (B(κ)). Questions for future research. Question 4.1. Is the function κ 7→ dimH (B(κ)) continuous for all m, n and for all norms k · kµ , k · kν ? Question 4.2. Given m, n and k · kµ , k · kν , what is the smallest κ > 0 such that dimH (B(κ)) = 0?

5. R ANDOM WALKS ON HOMOGENEOUS SPACES AND D IOPHANTINE APPROXIMATION ON FRACTALS Results about Diophantine approximation on fractals can be naturally divided into two classes: those demonstrating the largeness (in some sense) of the set of points on a given fractal that are difficult to approximate by rationals, and those demonstrating the largeness of the set of points that are easy to approximate by rationals. A result of the first type tends to be easier to prove than a result of the second type. One reason for this is that it may happen that the fractal does not contain any rational points, in which case it is difficult to find points on the fractal that are well approximated by rationals. In particular, the set of badly approximable matrices is known to have full Hausdorff dimension on a large class of fractals, see [48, 50, 19, 18], whereas analogous results regarding the complement of the set of badly approximable matrices (i.e. the set of well approximable matrices) are missing. In fact, any such analogous result must be fairly limited in scope, since the set FN consisting of all numbers in (0, 1) whose continued fraction partial quotients are all ≤ N is very nice from a geometric point of view, but consists entirely of badly approximable numbers.

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In joint work with Barak Weiss [64], I proved such an analogous result for the class of limit sets of algebraic similarity iterated function systems. Here, an algebraic similarity iterated function system on the space M of m × n matrices is a finite collection of contractions of the form M 3 α 7→ λβαγ + δ ∈ M, where λ > 0, β (resp. γ) is an m × m (resp. n × n) orthogonal matrix, and δ ∈ M. We proved that if K is the limit set of an algebraic similarity iterated function system and µ is a Bernoulli measure on K, then µ-almost every matrix is well approximable. For example, almost every number on the middle-thirds Cantor set is well approximable. This special case had been proven earlier by Einsiedler, Fishman, and Shapiro [25], but their technique was specific to ×b-invariant sets in dimension 1. To prove our theorem, we extend the seminal results of Benoist and Quint [5, 6, 7] regarding random walks on homogeneous spaces by weakening one of their assumptions. Namely, they consider a semisimple algebraic group G, a lattice Γ ⊆ G, and a random walk on the homogeneous space X = G/Γ generated by a measure µ on G such that the group H generated by the support of µ is Zariski dense in G. We replace the assumption that H is Zariski dense by an assumption that the measure µ is “expanding” in an appropriate sense. To apply our extension of Benoist–Quint’s results to Diophantine approximation, we use the Dani correspondence principle, which relates the Diophantine properties of a matrix α ∈ M to the dynamical properties of the trajectory (at uα Zm+n )t≥0 in the homogeneous space X = G/Γ, where G = SLm+n (R), " et/m Im at =

Γ = SLm+n (Z), " # Im −α uα = . In

# e−t/n In

,

Namely, we show that if α is a random point with respect to a Bernoulli measure µ on the limit set K of an algebraic similarity iterated function system, then the trajectory (at uα Zm+n )t≥0 can be approximated by the trajectory of a certain random walk on X. Our extension of Benoist–Quint’s results then shows that this trajectory equidistributes in X and thus so does the trajectory (at uα Zm+n )t≥0 . According to the Dani correspondence principle, this implies that α is well approximable. Questions for future research. Question 5.1. Let K be the limit set of an algebraic similarity iterated function system (e.g. the ternary Cantor set), and let µ be the Gibbs measure of a H¨ older continuous potential function on K (not necessarily a Bernoulli measure). Does µ necessarily give full measure to the set of well approximable matrices?

R EFERENCES 1. Jinpeng An, Anish Ghosh, and Lifan Guan, Bounded orbits of diagonalizable flows on finite volume quotients of products of SL2 (R), https: //arxiv.org/abs/1611.06470, preprint 2016. (Cited on page 3.) 2. Jinpeng An, Lifan Guan, and Dmitry Kleinbock, Bounded orbits of diagonalizable flows on SL3 (R)/SL3 (Z), Int. Math. Res. Not. IMRN (2015), no. 24, 13623–13652. MR 3436158 (Cited on page 3.) 3. Dzmitry Badziahin and Stephen Harrap, Cantor-winning sets and their applications, http://arxiv.org/abs/1503.04738, preprint 2015. (Cited on page 3.) 4. Lu´ıs Barreira, Dimension theory of hyperbolic flows, Springer Monographs in Mathematics, Springer, Cham, 2013. MR 3087567 (Cited on page 6.) 5. Yves Benoist and Jean-Franc¸ois Quint, Mesures stationnaires et ferm´es invariants des espaces homog`enes (Stationary measures and invariant subsets of homogeneous spaces), Ann. of Math. (2) 174 (2011), no. 2, 1111–1162 (French). (Cited on page 8.)

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, Stationary measures and invariant subsets of homogeneous spaces (II), J. Amer. Math. Soc. 26 (2013), no. 3, 659–734. (Cited on page 8.)

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, Stationary measures and invariant subsets of homogeneous spaces (III), Ann. of Math. (2) 178 (2013), no. 3, 1017–1059. MR 3092475 (Cited on page 8.)

8. Victor Beresnevich, Anish Ghosh, David Simmons, and Sanju Velani, Diophantine approximation in Kleinian groups: singular, extremal, and bad limit points, http://arxiv.org/abs/1610.05964, preprint 2016. (Cited on page 2.) 9. Victor Beresnevich, Felipe Ram´ırez, and Sanju Velani, Metric Diophantine approximation: aspects of recent work, https://arxiv.org/ abs/1601.01948, preprint 2016. (Cited on page 3.) 10. Ryan Broderick, Lior Fishman, Dmitry Kleinbock, Asaf Reich, and Barak Weiss, The set of badly approximable vectors is strongly C 1 incompressible, Math. Proc. Cambridge Philos. Soc. 153 (2012), no. 02, 319–339. (Cited on page 3.) 11. Ryan Broderick, Lior Fishman, and David Simmons, Badly approximable systems of affine forms and incompressibility on fractals, J. Number Theory 133 (2013), no. 7, 2186–2205. (Cited on page 3.) 12. Ryan Broderick, Lior Fishman, and David Simmons, Quantitative results using variants of Schmidt’s game: Dimension bounds, arithmetic progressions, and more, https://arxiv.org/abs/1703.09015, preprint 2017. (Cited on pages 2, 3, and 5.) 13. Ryan Broderick and Dmitry Kleinbock, Dimension estimates for sets of uniformly badly approximable systems of linear forms, Int. J. Number Theory 11 (2015), no. 7, 2037–2054. MR 3440444 (Cited on page 7.) 14. Jianyu Chen and Yakov Pesin, Dimension of non-conformal repellers: a survey, Nonlinearity 23 (2010), no. 4, R93–R114. MR 2602012 (Cited on page 6.) 15. Shrikrishna Gopal Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math. 359 (1985), 55–89. (Cited on page 2.) 16. Tushar Das, Lior Fishman, David Simmons, and Mariusz Urba´ nski, Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures, Selecta Math. (2017), 1–42. (Cited on page 2.) 17.

, Extremality and dynamically defined measures, part II: Measures from conformal dynamical systems, http://arxiv.org/abs/ 1508.05592, preprint 2015. (Cited on page 2.)

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, Badly approximable points on self-affine sponges and the lower Assouad dimension, http://arxiv.org/abs/1608.03225, preprint 2016, to appear in Ergodic Theory Dynam. Systems. (Cited on pages 2 and 7.)

19.

, Badly approximable vectors and fractals defined by conformal dynamical systems, http://arxiv.org/abs/1603.01467, preprint 2016, to appear in Math. Res. Lett. (Cited on pages 2 and 7.)

20. Tushar Das, Lior Fishman, David Simmons, and Mariusz Urba´ nski, A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation, https://arxiv.org/abs/1704.05277, preprint 2017, announcement. (Cited on pages 2 and 3.) 21. Tushar Das and David Simmons, The Hausdorff and dynamical dimensions of self-affine sponges: a dimension gap result, Invent. Math. (2017), 1–50. (Cited on pages 2, 5, and 6.) 22. Tushar Das, David Simmons, and Mariusz Urba´ nski, Tukia’s isomorphism theorem in CAT(-1) spaces, Ann. Acad. Sci. Fenn. Math. 41 (2016), 659–680. (Cited on page 2.) 23. 24.

, Dimension rigidity in conformal structures, Adv. Math. 308 (2017), 1127–1186. (Cited on page 2.) , Geometry and dynamics in Gromov hyperbolic metric spaces: with an emphasis on non-proper settings, http://arxiv.org/abs/ 1409.2155, preprint 2014, to appear in Math. Surveys Monogr. (Cited on page 2.)

25. Manfred Einsiedler, Lior Fishman, and Uri Shapira, Diophantine approximations on fractals, Geom. Funct. Anal. 21 (2011), no. 1, 14–35. (Cited on page 8.) 26. Kenneth Falconer, Fractal geometry: Mathematical foundations and applications, John Wiley & Sons, Ltd., Chichester, 1990. (Cited on page 3.) 27. Lior Fishman, Schmidt’s game on fractals, Israel J. Math. 171 (2009), no. 1, 77–92. (Cited on page 3.) 28. Lior Fishman, Dmitry Kleinbock, Keith Merrill, and David Simmons, Intrinsic Diophantine approximation on manifolds, http://arxiv. org/abs/1405.7650v2, preprint 2014. (Cited on page 2.) 29.

, Intrinsic Diophantine approximation on manifolds: General theory, http://arxiv.org/abs/1509.05439, preprint 2015, to appear in Trans. Amer. Math. Soc. (Cited on pages 2 and 3.)

30. Lior Fishman, Bill Mance, David Simmons, and Mariusz Urba´ nski, Shrinking targets for non-autonomous dynamical systems corresponding to Cantor series expansions, Bull. Aust. Math. Soc. 92 (2015), no. 2, 205–213. (Cited on page 2.) 31. Lior Fishman, Keith Merrill, and David Simmons, Hausdorff dimensions of very well intrinsically approximable subsets of quadratic hypersurfaces, http://arxiv.org/abs/1502.07648, preprint 2015. (Cited on page 2.) 32.

, Uniformly de Bruijn sequences and symbolic Diophantine approximation on fractals, http://arxiv.org/abs/1605.07953, preprint 2016, to appear in Ann. Comb. (Cited on page 2.)

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33. Lior Fishman, Vanessa Reams, and David Simmons, The Banach-Mazur-Schmidt and Banach-Mazur-McMullen games, J. Number Theory 167 (2016), 169–179. MR 3504042 (Cited on pages 2 and 3.) 34. Lior Fishman and David Simmons, Intrinsic approximation for fractals defined by rational iterated function systems - Mahler’s research suggestion, Proc. Lond. Math. Soc. (3) 109 (2014), no. 1, 189–212. (Cited on page 2.) 35.

, Extrinsic Diophantine approximation on manifolds and fractals, J. Math. Pures Appl. (9) 104 (2015), no. 1, 83–101. (Cited on page 2.)

36.

, Unconventional height functions in simultaneous Diophantine approximation, Monatsh. Math. 182 (2017), no. 3, 577–618. MR 3607503 (Cited on page 1.)

37. Lior Fishman, David Simmons, and Mariusz Urba´ nski, Diophantine approximation in Banach spaces, J. Th´eor. Nombres Bordeaux 26 (2014), no. 2, 363–384. (Cited on page 1.) 38. 39.

, Diophantine properties of measures invariant with respect to the Gauss map, J. Anal. Math. 122 (2014), 289–315. (Cited on page 2.) , Rigidity of limit sets for nonplanar geometrically finite Kleinian groups of the second kind, Geom. Dedicata 178 (2015), 95–101. (Cited on page 2.)

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