DIOPHANTINE APPROXIMATION ON TRANSLATION SURFACES

Transcription

1 JAYADEV S. ATHREYA UNIVERSITY OF WASHINGTON GOA, FEBRUARY 2016 DIOPHANTINE APPROXIMATION ON TRANSLATION SURFACES

2 THE ORIGINAL, AND STILL CLASSIC 2 R,p,q 2 Z p q < (q) Classical Diophantine Approximation

3 GEOMETRY OF NUMBERS LATTICES For a geometer, this is a statement about the geometry of (a family of) lattices. In particular, this is a family of shears of the standard integer lattice. = Z 2

4 SHEARING

5 APPROXIMATING ALPHA A typical vector in has the form p q q So asking to solve the inequality p q < (q) Is the same as finding lattice points x y 2 such that x <y (y)

6 1/31/2016 y = 1/x CLASSICAL EXAMPLE (y) = 1 y 2 PIGEONHOLE PRINCIPLE

7 SOME CLASSICAL QUESTIONS APPROXIMATIONS AND PROBABILITY Fix, vary Fix, vary (y) =A/y (y) =A/N infinitely many solutions infinitely many solutions solutions in range cn < q< dn solutions in range cn < q < dn measure/hdim of set of alpha diophantine exponent of alpha limit measure as N grows limit measure as N grows

8 HOW DO WE GENERALIZE? ALSO, HOW DO WE SOLVE?

9 DISCRETE SETS HIGHER DIMENSION/GENUS A lattice yields a flat torus. Higher-dimensional Diophantine approximation, related to higherdimensional lattices (or flat tori). Higher genus (flat) surfaces. C/ A flat torus is a parallelogram with parallel sides identified by translation. A translation surface is a general Euclidean polygon with parallel sides identified by translation.

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12 TRANSLATION SURFACES Translation surfaces of genus at least 2 have isolated cone type singularities, with angles integer multiples of 2. The order of a singularity is a measure of the excess angle. A singular point has order k if the angle is. 2 (k + 1) In complex analytic terms, we obtain a Riemann surface X and a holomorphic one-form w. Singular points of order k are zeros of w of order k.

13 SADDLE CONNECTIONS a saddle connection is a straight line trajectory connecting two zeros 2 HOLONOMY VECTORS Z To each saddle connection, we associate a holonomy vector 7!! 2 C 1 The set of holonomy vectors of the surface 1 p 2 1 (X, w) is denoted!

14 THE SET OF HOLONOMY VECTORS IS DISCRETE

15 SL(2, R) ACTS ON THE SPACE OF TRANSLATION SURFACES LINEAR ACTION ON POLYGONS ERGODIC ABSOLUTELY CONTINUOUS INVARIANT MEASURE HOLONOMY VECTORS VARY EQUIVARIANTLY g! = g!

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17 GROWTH SADDLE CONNECTION HOLONOMIES HAVE QUADRATIC GROWTH. Masur, Veech, Eskin-Masur, Vorobets

18 DIOPHANTINE PROBLEMS UNDERSTANDING SADDLE CONNECTIONS How well does this set approximate lines? Are there analogues of classical Diophantine results? What is the (fine scale) distribution of directions?

19 REVISITING THE CLASSICAL SETTING A typical vector in So asking to solve the inequality p q has the form q p q apple q Is the same as finding lattice points x y 2 such that x < y ( 1) The diophantine exponent is the supremum of values with infinitely many solutions.

20 DIOPHANTINE EXPONENT OF A TRANSLATION SURFACE sup : x y 2! such that x < y ( 1) has infinitely many solutions := µ(!) HOW WELL CAN YOU APPROXIMATE THE VERTICAL DIRECTION?

21 AXIOMATIC SET-UP Space X (moduli space of lattices, translation surfaces), SL(2, R) equivariant assignment of discrete set x in the plane. For x in X, `(x) =min{kvk : v 2 x } (x) = max{kvk 1 : v 2 x }

22 g t = e t 0 0 e t g t = e t 0 0 e t 1 0 h s = s 1

23 AXIOMATIC THEOREM For translation surfaces, lattices, and other discrete equivariant assignments EXPONENTS AND ORBITS lim sup t!1 lim sup s!1 log (g t x) t log (h s x) log s =1 = µ(x) 1 µ(x) CUSP EXCURSIONS ON PARAMETER SPACES, JLMS

24 LINEAR ALGEBRA WHEN DO VECTORS GET SHORT? A geodesic orbit of a vector is shortest when the components of the vector become equal. A horocycle orbit of a vector is shortest when the vector becomes horizontal. The arguments generalize to higher dimensions, subspaces, matrices, etc. Measure estimates + Borel-Cantelli imply that for almost every translation surface, the diophantine exponent is 2.

25 DISTRIBUTION OF APPROXIMATES Given an object x, what is the distribution of vectors v in the associate discrete set satisfying v 1 v 2 <A N< v 2 <cn Here, A>0, c>1.

26 PROBABILISTIC APPROXIMATION Special case: suppose h 1 x = x Consider the measure { :#( h x \ R A,c,N )=k} R A,c,N = (v 1,v 2 ) 2 R 2 : v 1 v 2 < A, N < v 2 <CN Does this have a limit as N grows?

27 ERDOS-SZUSZ-TURAN, KESTEN-SOS K=0 STUDIED EXTENSIVELY PAUL TURAN AND VERA SOS

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29 RENORMALIZATION TRANSLATES OF HOROCYCLES g log N R A,c,N = R A,c,1 #( h x \ R A,c,N )= #(g log N h xr A,c,1 ) educes to limiting distribution of d on {g log N h x :0apple apple 1} in X

30 IN MANY SETTINGS, THESE LONG CLOSED HOROCYCLES EQUIDISTRIBUTE WITH RESPECT TO SOME NATURAL MEASURE. FOR TRANSLATION SURFACES, UNDERSTANDING THE LIMIT MEASURE IS A DEEP QUESTION. FOR SPECIAL (LATTICE) SURFACES, REDUCES TO THE HYEPRBOLIC SETTING.

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32 GENERAL SOLUTION GENERALIZATION TO EQUIVARIANT PROCESSES, APPLICATIONS TO DIOPHANTINE PROBLEMS IN MANY CONTEXTS A.-Ghosh-Taha, forthcoming

33 THE FUTURE QUESTIONS/CONJECTURES Khintchin s theorem for translation surfaces (shrinking target property for geodesic flow on space of translation surfaces). Duffin-Schaefer for translation surfaces. Applications to approximation in algebraic number fields.