QUANTITATIVE DISTRIBUTIONAL ASPECTS OF GENERIC DIAGONAL FORMS Quantitative Oppenheim theorem Eigenvalue spacing for rectangular billiards Pair correlation for higher degree diagonal forms
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
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
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
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
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 <.
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.
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
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, 086... infinitely often 9
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
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.
PAIR CORRELATION FOR HOMOGENOUS FORMS OF DEGREE k Let F (x,..., x k ) = x k + α 2x k 2 + + 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.
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
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)
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
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
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
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
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.
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
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
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
The lower bound log(e log 2) Proposition For almost all α, letting c = log 2 = 0.086... 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
PAIR CORRELATION FOR HOMOGENOUS DIAGONAL FORMS Recall that and R(a, b; T ) = T F (x,..., x k ) = x k + α 2x k 2 + + 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
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
The following question seems open. Problem It is true that for all α R inf (m + n) mnα = 0? m,n Z + 26