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Equidistribution of Curves in Homogeneous Spaces and Dirichlet’S Approximation Theorem for Matrices

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Equidistribution of Curves in Homogeneous Spaces and Dirichlet’S Approximation Theorem for Matrices Manuscript submitted to doi:10.3934/xx.xx.xx.xx AIMS' Journals Volume X, Number 0X, XX 200X pp. X{XX EQUIDISTRIBUTION OF CURVES IN HOMOGENEOUS SPACES AND DIRICHLET'S APPROXIMATION THEOREM FOR MATRICES Nimish Shah Department of Mathematics, The Ohio State University Columbus, OH 43210, USA Lei Yang∗ College of Mathematics, Sichuan University Chengdu, Sichuan, 610065, China (Communicated by Shouchuan Hu) Abstract. In this paper, we study an analytic curve ' : I = [a; b] ! M(m × n; R) in the space of m by n real matrices, and show that if ' satisfies certain geometric condition, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem can not be improved. To do this, we embed the curve into a homogeneous space G=Γ, and prove that under the action of some expanding diagonal subgroup A = fa(t): t 2 Rg, the translates of the curve tend to be equidistributed in G=Γ, as t ! +1. The proof relies on the linearization technique and representation theory. 1. INTRODUCTION. 1.1. Diophantine approximation for matrices. In 1842, Dirichlet proved the following result on simultaneous approximation of a matrix of real numbers by integral vectors: Given two positive integers m and n, a matrix Φ 2 M(m × n; R), and any N > 0, there exist integral vectors p 2 Zn n f0g and q 2 Zm such that kpk ≤ N m and kΦp − qk ≤ N −n; (1.1) where k·k denotes the supremum norm; that is, kxk := max1≤i≤kjxij for x = k (x1; x2; : : : ; xk) 2 R . Now we consider the following finer question: for a particular m by n matrix Φ, could we improve Dirichlet's Theorem? By improving Dirichlet's Theorem, we mean that there exists a constant 0 < µ < 1, such that for all large N > 0, there exists nonzero integer vector p 2 Zn with kpk ≤ µN m, and integer vector q 2 Zm such E-mail address: [email protected], [email protected]. 2010 Mathematics Subject Classification. Primary: 11J13, 11J83; Secondary: 22E40, 37A17. Key words and phrases. Dirichlet's theorem, Diophantine approximation, homogeneous spaces, equidistribution, Ratner's theorem. This material is based upon work supported by the National Science Foundation under Grant Nos. 1301715 and 1700394. The second author is supported in part by ISF grant 2095/15, ERC grant AdG 267259, NSFC grant 11743006, 11801384 and the Fundamental Research Funds for the Central Universities YJ201769. ∗ Corresponding author: Lei Yang. 1 −n that kΦp−qk ≤ µN . If such constant µ exists, then we say Φ is DTµ-improvable. If Φ is DTµ-improvable for some 0 < µ < 1, then we say Φ is DT -improvable (here DT stands for Dirichlet's Theorem). This problem was first studied by Davenport and Schmidt in [7], in which they proved that almost every matrix Φ 2 M(m × n; R) is not DT -improvable. In [7], they also proved the following result. For m = 1 and n = 2, M(1 × 2; R) = R2, one considers the curve φ(s) = (s; s2) in R2. Then for almost every s 2 R with respect to the Lebesgue measure on R, φ(s) is not DT1=4 improvable. This result was generalized by Baker in [3]: for any smooth curve in R2 satisfying some curvature condition, almost every point on the curve is not DTµ improvable for some 0 < µ < 1 depending on the curve. Bugeaud [4] generalized the result of Davenport and Schmidt in the following sense: for m = 1, and general n, almost every point 2 n on the curve '(s) = (s; s ; : : : ; s ) is not DTµ-improvable for some small constant 0 < µ < 1. Their proofs are based on the technique of regular systems introduced in [7]. Recently, based on an observation of Dani [5], as well as Kleinbock and Margulis [11], Kleinbock and Weiss [10] studied this Diophantine approximation problem in the language of homogeneous dynamics, and proved the following result: for m = 1 and arbitrary n, if an analytic curve in M(1 × n; R) =∼ Rn is not contained in any proper affine subspace, then almost every point on the curve is not DTµ-improvable for some small constant 0 < µ < 1 depending on the curve. Based on the same correspondence, Nimish Shah [18] proved the following stronger result: for m = 1 and general n, if an analytic curve ' : I = [a; b] ! Rn is not contained in a proper affine subspace, then almost every point on the curve is not DT -improvable. For m = n, Lei Yang [20] provided a geometric condition and proved that if an analytic curve ' : I = [a; b] ! M(n × n; R) satisfies the condition, then almost every point on ' is not DT -improvable. The geometric condition given there provides some hint on solving the problem for general (m; n), and will be discussed in detail later. The purpose of this paper is to give a geometric condition for each (m; n), and show that if an analytic curve ' : I = [a; b] ! M(m × n; R) satisfies the condition, then almost every point on ' is not DT -improvable. The geometric conditions called generic condition and supergeneric condition are defined as follows: Definition 1.1. For any m and n, let ' : I = [a; b] ! M(m × n; R) denote an analytic curve. For m = n, we say ' is generic at s0 2 I if there exists a subinterval Js0 ⊂ I such that for any s 2 Js0 n fs0g, '(s) − '(s0) is invertible. In order to define supergeneric condition, we need additional notation. We consider the following two embeddings from M(m × m; R) to the Lie algebra sl(2m; R) of SL(2m; R): 0 X n+ : X 2 M(m × m; ) 7! 2 sl(2m; ); R 0 R and 0 n− : X 2 M(m × m; ) 7! 2 sl(2m; ): R X 0 R 2 Let Im Em := 2 sl(2m; R): (1.2) −Im + − −1 If X 2 M(m × m; R) is invertible, then the triple fn (X); Em; n (X )g forms a standard basis of a copy of sl(2; R). A Lie subgroup L of H = SL(2m; R) is called observable if there exists a finite dimensional linear representation V of H and a nonzero vector v 2 V such that the subgroup of H stabilizing v is equal to L. A Lie subalgebra l is called an observable Lie subalgebra of sl(2m; R), if it is the Lie algebra of some observable Lie subgroup L ⊂ H. Further we are interested in an observable subalgebra l containing Em. Let ± ± L = fX 2 M(m × m; R): n (X) 2 lg: ± ± If X 2 L is invertible, then n (X)v = 0 and Emv = 0. Therefore by the basic ∓ −1 −1 ∓ property of sl2-triples, we get n (X )v = 0. Therefore X 2 L . In particular, if L− = M(m × m; R), then l = sl(2m; R). For the case of m = n, the curve ' is called supergeneric at s0 2 I if it is generic at s0 (with subinterval Js0 ⊂ I), and for any proper observable subalgebra l of sl(2m; R) containing Em, we have −1 −1 − f('(s1) − '(s0)) − ('(s2) − '(s0)) ): s1; s2 2 Js0 n fs0gg 6⊂ L : (1.3) When m 6= n, we will define what it means to say that ' is generic (supergeneric) at s0 2 I by induction on m + n as follows: For m < n, we express '(s) = ['1(s); '2(s)], where '1(s) is the first m by m block, and '2(s) is the rest m by n − m block. We say ' is generic (supergeneric) at s0 2 I, if there exists a subinterval Js0 ⊂ I such that for any s 2 Js0 n fs0g, '1(s) − '1(s0) is invertible; and if we define : Js0 n fs0g ! M(m × (n − m); R), by −1 (s) := ('1(s) − '1(s0)) ('2(s) − '2(s0)) then is generic (supergeneric) at some s1 2 Js0 n fs0g. For m > n, ' is called generic (supergeneric) at s0 2 I if its transpose T ' : I = [a; b] ! M(n × m; R) is generic (supergeneric) at s0. We say that ' is generic (supergeneric) or satisfies generic (supergeneric) con- dition, if ' is generic (supergeneric) at some s0 2 I. Since ' is analytic, if it is generic (supergeneric) at one point of I then it will be generic (supergeneric) at all but finitely many points of I. Remark 1.2. We will discuss the generic condition and the supergeneric condition in detail in AppendixA. Here we list several important statements. 1. If m and n are coprime, then the generic condition is the same as the super- generic condition (see Proposition A.1). 2. If m = 1 or n = 1, the generic condition (which is the same as the supergeneric condition) is equivalent to the condition that the curve is not contained in any proper affine subspace (see Proposition A.2). 3 3. In [20], it is proved that for m = n, if there exists s0 2 I and a subinterval 0 Js0 ⊂ I such that the derivative ' (s0) is invertible, '(s) − '(s0) is invertible for any s 2 Js0 n fs0g, and −1 f('(s) − '(s0)) : s 2 Js0 n fs0gg is not contained in any proper affine subspace of M(m × m; R), then almost every point on the curve is not DT -improvable. Clearly this condition implies the supergeneric condition. 4. For any m and n, the set of supergeneric curves in M(m × n; R) is open and dense in the set of analytic curves in M(m × n; R) (see Proposition A.6).
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