Quantitative Oppenheim theorem. Eigenvalue spacing for rectangular billiards. Pair correlation for higher degree diagonal forms

Transcription

1 QUANTITATIVE DISTRIBUTIONAL ASPECTS OF GENERIC DIAGONAL FORMS Quantitative Oppenheim theorem Eigenvalue spacing for rectangular billiards Pair correlation for higher degree diagonal forms

2 EFFECTIVE ESTIMATES ON INDEFINITE TERNARY FORMS Theorem (Dani Margulis) Let Q be an indefinite, irrational, ternary quadratic form. Then the set {Q(v) : v Z 3 primitive} is dense in the real line. There is the following quantification obtained by an effective version of Ratner s theorem. 2

3 Theorem (Lindenstrauss Margulis) Let Q be a given indefinite, ternary quadratic form with det Q = and ε > 0. Then, for T large enough, at least one of the following holds There is an integral quadratic form Q such that det Q < T ε and Q λq T where λ = det Q /3 For any ξ [ (log T ) c, (log T ) c], there is primitive v Z 3, 0 < v < T C such that Q(v) ξ (log T ) c 3

4 Let Q be as above. For which functions A(N) and δ(n) 0 (depending on Q) does the statement max min ξ <A(N) x Z 3 0< x <N Q(x) ξ < δ(n) hold? In the [L-M] result, A(N) and δ(n) depend logarithmically on N. Consider diagonal forms of signature (2, ) Q(x) = x 2 + α 2x 2 2 α 3x 2 3 (α 2, α 3 > 0) There is the stronger statement for one-parameter families ( ) Theorem (B) Let α 2 > 0 be fixed. For most α 3 > 0, the following holds (i) Assuming the Lindelöf hypothesis min x Z 3,0< x <N (ii) Unconditionally, Q(x) N +ε and ( ) holds provided A(N)δ(N) 2 N ε min x Z 3,0< x <N Q(x) N 2 5 +ε and ( ) holds assuming A(N) 3 δ(n) 2 N ε

5 REMARKS The statement min x Z 3,0< x <N Q(x) N +ε is essentially best possible For generic diagonal forms, i.e. considering both α 2 and α 3 as parameters, (i) holds without need of Lindelöf Ghosh and Kelmer obtained similar results for generic elements in the full space of indefinite ternary quadratic forms 5

6 EIGENVALUE SPACINGS OF FLAT TORI Berry Tabor Conjecture: local statistics of generic integrable quantum Hamiltonian coincide with those of random numbers generated by Poisson process PAIR CORRELATION FOR 2D FLAT TORI Γ R 2 lattice and M = R 2 /Γ associated flat torus. Eigenvalues of M given by 4π 2 v 2, v Γ (= dual lattice). These are the values at integral points of binary quadratic form B(m, n) = 4π 2 n v + n 2 v 2 2, with {v, v 2 } a Z-basis for Γ. Weyl law holds {j : λ j (M) T } c M T with c M = (area M )/4π Distribution of local spacings λ j λ k, j k R(a, b, T ) = { (j, k) : j k, λj T, λ k T, a λ j λ k b } T where < a < b <.

7 According to the Berry Tabor conjecture, λ j λ k (j k) should be uniformly distributed on R. Thus for given a < b, lim R(a, b, T ) = T c2 M (b a) ( ) Theorem (Sarnak, 996) space of all tori. ( ) holds for a set of full measure in the Analytical approach which seems to require 2 parameters and not applicable to rectangular billiards with eigenvalues αm 2 + n 2 (where α is the aspect ratio between height π and width π/ α). Theorem (Eskin-Margulis-Mozes) ( ) holds for (weakly) diophantine billiards, assuming 0 (a, b) In particular ( ) is valid for rectangular billiards provided α is diophantine, in the sense that qα > q C for some C > 0. Ergodic theory approach with no (or very poor) quantitative versions if we let b a 0 with T.

8 MINIMAL GAP FUNCTION OF SPECTRUM OF RECTANGULAR BILLIARD For α irrational, we get simple spectrum 0 < λ < λ 2 <, with growth {j : λ j T } = {(m, n); m, n, αm 2 + n 2 T } π 4 α T Definition δ (α) min (N) = min(λ i+ λ i : i N) For a Poisson sequence of N uncorrelated levels with unit mean spacing, the smallest gap is almost surely of size N. 8

9 Remark For the Gaussian unitary ensemble, on the scale of the mean spacing, smallest eigenphase gap is N /3, while for the Gaussian orthogonal ensemble, it is expected that the minimal gap is of size N /2. Theorem (Bloomer B Radziwill Rudnick) For almost all α > 0 δ (α) min (N) N ε δ (α) (log N)c min (N) N with c = log(e log 2) log 2 = 0, infinitely often 9

10 PAIR CORRELATION FOR GENERIC 3D RECTANGULAR BILLIARDS Consider generic positive definite ternary quadratic form Q(x) = x 2 + αx2 2 + βx2 3 with α, β > 0. Restricting the variables to Z [0, N], mean spacing is O( N ). For T large and a < b, denote R(a, b; T ) = T 3/2 { (m, n) Z 3 + Z 3 + : m n, Q(m) T, Q(n) T and Q(m) Q(n) [a, b] } and c = lim ε 0 ε { (x, y) R 3 + R3 + ; Q(x) and Q(x) Q(y) < ε 2} 0

11 Theorem (B) Let Q = Q α,β with α, β > 0 parameters. Almost surely in α, β, the following holds. Fix 0 < ρ <. Let T and a < b, a, b < O() such that T ρ < b a <. Then R(a, b; T ) ct 2(b a) Remark Vanderkam established the pair correlation conjecture for generic four-dimensional flat tori using an unfolding procedure.

12 PAIR CORRELATION FOR HOMOGENOUS FORMS OF DEGREE k Let F (x,..., x k ) = x k + α 2x k α kx k k with α 2,..., α k > 0. Thus average spacing between values of P is asymptotically constant. Define R(a, b; T ) = T { (m, n) Z k + Zk + : m n, F (m) T, F (n) T and F (m) F (n) [a, b]} Theorem (B) Consider α 2,..., α k as parameters. There is ρ > 0 such that almost surely in α 2,..., α k the following statement holds. Let T and a, b = O(), b a > T ρ. Then R(a, b; T ) c 2 (b a) where c = {x R k + : F (x) } Results of this type for generic elements in the full space of homogenous forms were obtained by Vanderkam (999) using Sarnak s method.

13 QUANTITATIVE OPPENHEIM FOR -PARAMETER FAMILIES Harmonic analysis approach but rather than expressing the problem using Gauss sums, reformulate it multiplicatively and rely on Dirichlet sum behavior. For instance, the sum n N is small for say t > N ε ; under the Lindelöf hypothesis one gets n N n it n it N 2( + t ) ε This permits to decompose in small and large t, where the small-t range provides main term and large-t contribution is an error term. 3

14 Consider one-parameter family Q α (x) = x 2 + x2 2 αx2 3 with α > 0 a parameter Taking x, x 2, x 3 N, condition x 2 + x2 2 αx2 3 < δ is rewritten as x 2 + x2 2 αx 2 3 δ N 2 or log(x 2 + x 2 2 ) 2 log x 3 log α The number of x, x 2, x 3 N for which this inequality holds is then expressed using Fourier transform as T t <T F (t) F 2 (2t) e it log α dt with T = N 2 δ δ N 2 F (t) = F 2 (t) = x,x 2 N x N x it (x 2 + x2 2 )it (partial sum of Epstein zeta function) (partial sum of Riemann zeta function)

15 Split t <T = t <N 2 + N 2< t <T = () + (2) Contribution of () amounts to δ N 2 N 2 x,x 2,x 3 N [ log(x 2 +x 2 2 ) 2 log x 3 log α <N 2] δn without further assumptions on α To bound contribution of (2), average in α and use Parseval T [ N 2< t <T ] F (t) 2 F 2 (t) 2 2 ( dt T 2 max N 2< t <T F 2 (t) ) (. T t <T ) F (t) 2 2 dt 5

16 Assuming Lindelöf For second factor T F (t) 2 dt t <T { max F 2(t) N 2T ε N ε < t <T # (x, x 2, x 3, x 4 ) Z 4 ; x i N and log(x 2 + x2 2 ) log(x2 3 + x2 4 ) < T # {(x, x 2, x 3, x 4 ) Z 4 ; x i N and x 2 + x2 2 x2 3 x2 4 < N 2 } T = δ N 2+ε This gives ( δ 2 cδn + O N.N ) 2 +ε N +ε δn provided N +ε < δ } 6

17 Unconditionally, one may show that a.s. in α min x Z 3 \{0} x <N Q α (x) N 2 5 +ε by bounding error term using distributional estimates. Main ingredients Large value estimate for Dirichlet polynomials Lemma Define S(t) = n M a n n it where a n. Then for T > M mes [ t < T : S(t) > V ] M ε (M 2 V 2 + M 4 V 6 T ) (due to Jutila, 977) Applied here with M = N 2 and S(t) = F 2 (t) 2. 7

18 Bounds on partial sums of the Epstein zeta function Lemma For t > N 2, one has the estimate m,n N (m 2 + n 2 ) it N t 3 +ε Follows the steps of Van der Corput s third derivative estimate, i.e. ( 6, 2 3 ) is a (2-dim) exponent pair Appears in a paper of V. Blomer on Epstein zeta functions where it is used to establish a bound E ( 2 + it ) t 3 +ε where E(s) = x Z 2 x 0 Q(x) s 8

19 SMALL GAPS IN SPECTRUM OF RECTANGULAR BILLIARD Recall that {λ < λ 2 < } = {αm 2 + n 2 : m, n Z} δ (α) min (N) = min (λ i+ λ i ) i N In order to bound δ (α) min (N), consider equivalently (M = N) min { n 2 n2 2 + α(n2 3 n2 4 ) : n i < M and (n, n 3 ) (n 2, n 4 ) } or min { n n 2 αn 3 n 4 : 0 < n i < M } ( ) Problem Is it true that ( ) M 2 ε for all α? In [B-B-R-R], this was shown for certain quadratic irrationals and also generically Proposition ( ) M 2 ε for almost all α.

20 Taking n i M, condition is replaced by SKETCH OF THE PROOF n n 2 αn 3 n 4 < δ log n + log n 2 log n 3 log n 4 log α < δ M 2 = T This leads to evaluation of T F t <T (t) 4 e it(log α) dt with F (t) = n it Split again t <T = t <M 2 + M 2< t <T n M = () + (2) Contribution of () evaluated independently of α and gives cδm 2 Contribution of (2) estimated in the α-mean and by Parseval bounded by 20

21 T [ Assuming Lindelöf, one obtains M 2< t <T F (t) 8 dt ] 2 M 2< t <T F (t) 8 dt T +ε M 4 leading to the condition δm 2 > T 2 +ε M 2, hence δ > M 2+ε. The Lindelöf hypothesis may be avoided in the following way. Replace n i M by n i = n i n i with n i M σ and n i M σ (σ arbitrary small and fixed). We obtain where T F (t) = t <T F (t) 4 G(t) 4 e it(log α) dt n M σ n it and G(t) = n it n M σ 2

22 Error term becomes Estimate T M 2< t <T F (t) 8 G(t) 8 dt M 2< t <T F (t) 8 G(t) 8 dt ( max M 2< t <T 2 G(t) 8 ). t <T F (t)4 2 dt This gives < M 8σ( τ) (T + M 4( σ) ) M 4( σ) for some τ = τ(σ) > 0 { (T δm 2 ε + O 2 + T M 2( σ) = δm 2 ε + O {(δ 2M } +2σ + δ)m 2 4στ ) M 2+2σ 4στ } and condition δ > M 2+4σ 22

23 The lower bound log(e log 2) Proposition For almost all α, letting c = log 2 = δ (α) (log N)c min (N) N infinitively often uses K. Ford s result # { u.v : u, v Z +, u, v < N } 2 N(log N) c (log log N) 2/3 and shows small deviation from expected Poisson statistics for the eigenvalues. 23

24 PAIR CORRELATION FOR HOMOGENOUS DIAGONAL FORMS Recall that and R(a, b; T ) = T F (x,..., x k ) = x k + α 2x k α kx k k (α i > 0) { (m, n) Z k + Zk + ; m n, F (m) T, F (n) T and F (m) F (n) [a, b]} A first step consists in a localization of the variables m i I i, n i J i where I i, J i are intervals of size T k 2ε satisfying a separation condition dist (I i, J i ) > T k ε. Let ξ = a+b 2 and δ = b a 2. Evaluate { (m, n) Z k Z k ; m i I i, n i J i and F (m) F (n) ξ < δ } ( ) T Condition F (m) F (n) ξ < δ is replaced by ( log m k n k +α 2(m k 2 nk 2 )+ +α k (m k k nk k ) ξ) log(m k k nk k ) log α δ k < α k = B where m k k nk k for m k I k, n k J k

25 Evaluate ( ) by S (t)s 2 (t) e it log α k ˆ [ B, B ] (t)dt ( ) with S (t) = ( m k n k + α 2(m k 2 nk 2 ) + + α k (m k k nk k ) ξ) it S 2 (t) = m i I i,n i J i m I k,n J k (m k n k ) it Set B 0 = α k T κ and decompose [ B, B ] = B 0 B [ B, 0 B ] + ( 0 [ B, B ] B 0 B ) [ B, 0 B ] producing in ( ) the main contribution and an 0 error term. Evaluation of the error term uses pointwise bound on S 2 (t) and an estimate T t <T S (t) 2 dt T 3 4 k +ε in average over α 2,..., α k 25

26 The following question seems open. Problem It is true that for all α R inf (m + n) mnα = 0? m,n Z + 26