Fine-scale statistics in number theory, geometry and dynamics Jens Marklof University of Bristol http://www.maths.bristol.ac.uk Lattice Point Distribution and Homogeneous Dynamics ICERM, Brown University June 2020 Research supported by EPSRC https://people.maths.bris.ac.uk/~majm/bib/talks/ICERM_Marklof2020.pdf ; file version June 28, 2020 1
Here is the plan for our three sessions: • How can we measure randomness in deterministic sequences? • From deterministic sequences to random point processes • Case study 1: Hitting and return times for linear flows on flat tori • Case study 2: Fractional parts of √ n • [Case study 3: Directions in hyperbolic lattices] 2
How can we measure randomness in deterministic sequences 3
Gap statistics • Consider ordered sequence of real numbers 0 ≤ a 1 ≤ a 2 ≤ · · · → ∞ of density one , i.e., N [0 , T ] N [0 , T ] := # { n | a n ≤ T } . lim = 1 , T T →∞ This ensures the average gap between elements in this sequence is 1. • Gap distribution P T [ a, b ] = # { n ≤ N [0 , T ] | a n +1 − a n ∈ [ a, b ] } N [0 , T ] • The counting measure P T defines a probability measure on R ≥ 0 . Does P T converge (weakly) to some probability measure P as T → ∞ ? I.e., T →∞ P T [ a, b ] = P [ a, b ] lim ∀ 0 ≤ a < b < ∞ 4
Example: Integers For a n = n we have P T [ a, b ] = # { n ≤ N [0 , T ] | 1 ∈ [ a, b ] } = δ 1 [ a, b ] N [0 , T ] So P T = δ 1 = P (the Dirac mass at 1). 5
Example: Quadratic forms at integer lattice points I ∗ • Let ( a n ) n given by the set � � � π ( αm 2 + n 2 ) � � � m, n ∈ Z 2 4 √ α � ≥ 0 • Note ( a n ) n has density one (check!) w • We have no proof P T − → P with P the exponential distribu- tion for any α • For α ∈ Q one can show w P T − → δ 0 Note: The exponential distribution is the gap distribution (“waiting times”) of a Poisson point process of intensity one ∗ These examples are already discussed M. Berry and M. Taylor, Proc. Roy. Soc 1977 who where interested in energy level statistics in the context of quantum chaos 6
Example: Quadratic forms at integer lattice points II The previous example was a positive definite quadratic form. How about the following discriminant-zero case: • Let ( a n ) n given by the set � � � ( αm + n ) 2 � � � ( m, n ) ∈ Z 2 � ≥ 0 2 α • Note ( a n ) n has density one (check!) • One can show that P T does α ∈ / not converge for Q (only along subsequences), and understand the distribution in terms of the “three gap theo- rem” w Exercise 1: Show that for α ∈ Q we have P T − → δ 0 . 7
Example: Quadratic forms at integer lattice points II The previous example was a positive definite quadratic form. How about the following discriminant-zero case: • Let ( a n ) n given by the set � � � ( αm + n ) 2 � � � ( m, n ) ∈ Z 2 � ≥ 0 2 α • Note ( a n ) n has density one (check!) • One can show that P T does α ∈ / not converge for Q (only along subsequences), and understand the distribution in terms of the “three gap theo- rem” w Exercise 1: Show that for α ∈ Q we have P T − → δ 0 . 8
Example: Quadratic forms at integer lattice points II The previous example was a positive definite quadratic form. How about the following discriminant-zero case: • Let ( a n ) n given by the set � � � ( αm + n ) 2 � � � ( m, n ) ∈ Z 2 � ≥ 0 2 α • Note ( a n ) n has density one (check!) • One can show that P T does α ∈ / not converge for Q (only along subsequences), and understand the distribution in terms of the “three gap theo- rem” w Exercise 1: Show that for α ∈ Q we have P T − → δ 0 . 9
Rescaling Suppose the sequence 0 ≤ a 1 ≤ a 2 ≤ · · · → ∞ does not have density one, but satisfies the more general N [0 , T ] lim = 1 , N [0 , T ] := # { n | a n ≤ T } . L ( T ) T →∞ � T with the integrated density L ( T ) = ν [0 , T ] = 0 ν ( dt ) and the Borel mea- sure ν is absolutely continuous with respect to Lebesgue measure dt . Then the rescaled sequence b n = L ( a n ) has density one and it is more nat- ural consider the gap distribution for this rescaled sequence than the “raw” gap distribution the original sequence. � T � � Note N [0 , T ] = δ a n [0 , T ] = δ a n ( dt ) so we cannot take L ( T ) = 0 n n N [0 , T ] . Question: What would the gap distribution be for the choice L ( T ) = N [0 , T ] ? 10
Example: The Riemann zeros • Let a n be the imaginary part of the n th Riemann zero on the critical line (in the upper half plane). • Then a n has a density given by L ( T ) = T T 2 π log 2 π e . • Consider gap distribution of = the rescaled zeros b n 2 π log a n a n 2 π e . • We have no proof P T → P with P given by the limiting cap distri- bution for large unitary random matrices. 11
General fine-scale statistics • Consider 0 ≤ a 1 ≤ a 2 ≤ · · · → ∞ of density one (as before). • Fix σ a locally finite Borel measure on R ≥ 0 so that σ [0 , ∞ ) = ∞ . • For D ⊂ R a compact interval, set N ( D ) = # { n | a n ∈ D } and denote by t + D its translation by t . • For k ∈ Z ≥ 0 E σ ([0 , T ] , D, k ) = σ { t ∈ [0 , T ] | N [ t + D ] = k } σ [0 , T ] is the probability that, for t random in [0 , T ] (w.r.t. σ ), the interval t + D contains k contains elements of ( a n ) n 12
Example: Gap and nearest distance statistics ∞ � • Take D = [0 , s ] , k = 0 , σ = δ a n (so A = 1 by assumption); then n =1 E σ ([0 , T ] , [0 , s ] , 0) = # { n ≤ N [0 , T ] | N [ a n , a n + s ] = 0 } = P T [0 , s ] N [0 , T ] We have recovered the gap distribution of ( a n ) n ! • If instead we take D = [ − s, s ] , then E σ ([0 , T ] , [ − s, s ] , 0) = # { n ≤ N [0 , T ] | N [ a n − s, a n + s ] = 0 } N [0 , T ] which is the nearest distance distribution of ( a n ) n . 13
Example: Gap and void statistics • For σ = Leb (the Lebesgue measure normalised so that Leb[0 , 1] = 1 ) then E Leb ([0 , T ] , D, 0) = Leb { t ∈ [0 , T ] | N [ t + D ] = 0 } T is called the void distribution of ( a n ) n . Exercise 2 : Show that for T > 0 and s ∈ R > 0 \ { discont. } we have P T [ s, ∞ ) = − d N ( T ) dsE Leb ([0 , T ] , [0 , s ] , 0) T − 1 T ✶ ( a 1 > s ) − 1 T ✶ ( a N ( T ) > T − s ) { discont. } = { a 1 } ∪ { a n +1 − a n | n ≤ N ( T ) − 1 } ∪ { T − a N ( T ) } Hint: Work out E Leb ([ a n , a n +1 ) , [0 , s ] , 0) . 14
Example: Pigeon hole statistics ∞ � • Take D = [0 , s ) (e.g. s = 1 ), k ∈ Z ≥ 0 , σ = δ ns (so A = 1 /s ); then n =0 E σ ([0 , T ] , [0 , s ] , k ) = # { 0 ≤ n ≤ T/s | N [ ns, ( n + 1) s ) = k } ⌊ T/s ⌋ + 1 i.e. the proportion of bins [0 , s ] , [ s, 2 s ] , [2 s, 3 s ] , . . . , [ s ⌊ T/s ⌋ , s ( ⌊ T/s ⌋ + 1)] that contain exactly k points 15
Example: Quadratic forms at integer lattice points • Let ( a n ) n given by the set � � � π ( αm 2 + n 2 ) � � � m, n ∈ Z 2 4 √ α � ≥ 0 • In the experiment we have taken the pigeon hole stats E σ ([0 , T ] , [0 , s ] , k ) with bin width s = 3 . T →∞ E σ ([0 , T ] , [0 , s ] , k ) = s k k !e − s (the Poisson distribution), but • We expect lim no proof 16
Two-point correlations • The above local statistics are often too difficult to handle analytically; two- point statistics are more tractable • Pair correlation measure R T [ a, b ] = # { ( m, n ) | n ≤ N [0 , T ] , m � = n, a m − a n ∈ [ a, b ] } N [0 , T ] • Compare with gap distribution P T [ a, b ] = # { n ≤ N [0 , T ] | a n +1 − a n ∈ [ a, b ] } N [0 , T ] • For positive definite quadratic forms, under explicit Diophantine conditions on the coefficients, one can prove ∗ R T [ a, b ] → b − a ∗ A. Eskin, G. Margulis, S. Mozes, Annals Math 2005 17
Example: Riemann zeros A. M. Odlyzko, Math. Comp. 1987 best result to-date (and a beautiful paper) Z. Rudnick, P . Sarnak, Duke. Math. J. 1996 18
From deterministic sequences to random point processes 19
Randomization • Consider 0 ≤ a 1 ≤ a 2 ≤ · · · → ∞ of density one (as before). • Fix σ a locally finite Borel measure on R ≥ 0 so that σ [0 , ∞ ) = ∞ . • Let t be a random variable distributed on [0 , T ] with respect to σ ; that is t is defined by P ( t ∈ B ) = σ ( B ∩ [0 , T ]) for any Borel set B ⊂ R . [0 , T ] • Define the random point process (=a random counting measure on R ) ∞ � ξ T = δ a n − t n =1 • Note: E σ ([0 , T ] , D, k ) = P ( ξ T D = k ) . d • Is there a limiting point process ξ T − → ξ as T → ∞ ? 20
Point processes ∗ • M ( R ) the space of locally finite Borel measures on R , equipped with the vague topology † • N ( R ) ⊂ M ( R ) the closed subset of integer-valued measures, i.e., the set of ζ such that ζB ∈ Z ∪ {∞} for any Borel set B • A point process on R is a random measure in N ( R ) � • For ζ ∈ N ( R ) , we can write ζ = δ τ j ( ζ ) where τ j : N ( R ) → R . j • Use convention τ j ≤ τ j +1 , and τ 0 ≤ 0 < τ 1 if there are τ j ≤ 0 . • ζ is simple if sup ζ { t } ≤ 1 a.s t • The intensity measure of ζ is defined as E ζ . ∗ For general background see O. Kallenberg, Foundations of Modern Probability, Springer 2002 † The vague topology is the smallest topology such that the function � f : M ( R d ) → R , µ �→ µf is continuous for every f ∈ C c ( R d ) . 21
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