The two-point correlation function of the fractional parts of n is Poisson

Transcription

1 he two-point correlation function of the fractional parts of n is Poisson Daniel El-Baz Jens Marklof Ilya Vinogradov February, 04 Abstract Elkies and McMullen [Duke Math. J ] have shown that the gaps between the fractional parts of n for n =,..., N, have a limit distribution as N tends to infinity. he limit distribution is non-standard and differs distinctly from the exponential distribution expected for independent, uniformly distributed random variables on the unit interval. We complement this result by proving that the twopoint correlation function of the above sequence converges to a limit, which in fact coincides with the answer for independent random variables. We also establish the convergence of moments for the probability of finding r points in a randomly shifted interval of size /N. he key ingredient in the proofs is a non-divergence estimate for translates of certain non-linear horocycles. Introduction It is well known that, for every fixed 0 < α <, the fractional parts of n α n =,..., N are, in the limit of large N, uniformly distributed mod. Numerical experiments suggest that the gaps in this sequence converge to an exponential distribution as N, which is the distribution of waiting times in a Poisson process, cf. Fig.. he only known exception is the case α = /. Here Elkies and McMullen [] proved that the limit gap distribution exists and is given by a piecewise analytic function with a power-law tail Fig.. In the present study we show that a closely related local statistics, the two-point correlation function, has a limit which in fact is consistent with the Poisson process, see Fig. 3. he proof of this claim follows closely our discussion in [], which produced an analogous result for the two-point statistics of directions in an affine lattice. We note that Sinai [4] has recently proposed an alternative approach to the statistics of n mod, but will not exploit this here. Other number-theoretic sequences, whose two-point correlations are Poisson, include the values of positive definite quadratic forms subject to certain diophantine conditions he research leading to these results has received funding from the European esearch Council under the European Union s Seventh Framework Programme FP/ / EC Grant Agreement n J.M. is also supported by a oyal Society Wolfson esearch Merit Award. School of Mathematics, University of Bristol, Bristol BS8 W, U.K. MSC subject codes: primary J7; secondary K36, P, E40, 37A7, 37A5.

2 Figure : Gap distribution of the fractional parts of n /3 with n Figure : Gap distribution of the fractional parts of n with n 0 5.

3 Figure 3: wo-point correlations of the fractional parts of n with n 000, n /. [, 3], forms in more variables [6, 5, 7], inhomogeneous forms in two [8, 5] and more variables [7], and the fractional parts of n α and higher polynomials for almost all α [, 6, 4] specific examples, e.g. α = are still open. o describe our results, let us first note that n = 0 mod if and only if n is a perfect square. We will remove this trivial subsequence and consider the set P = { n mod : n, n / } := /Z. where N denotes the set of perfect squares. he cardinality of P is N =. We label the elements of P by α,..., α N. he pair correlation density of the α j is defined by N f = N f N α i α j + m,. N i,j= i j where f C 0 continuous with compact support. Note that N is not a probability density. Our first result establishes that N converges weakly to the two-point density of a Poisson process: heorem. For any f C 0, lim N f = fs ds..3 Both the convergence of the gap distribution and of the two-point correlations follow from a more general statistics, namely, the probability of finding r elements α j in randomly 3

4 placed intervals of size proportional to /N. Given a bounded interval I, define the subinterval J = J N I, α = N I + α + Z of length N I, and let N I, α = #P J N I, α..4 It is proved in [] that, for α uniformly distributed in with respect to the Lebesgue measure λ, the random variable N I, α has a limit distribution Ek, I. hat is to say, for every k Z 0, lim λ{α : N I, α = k} = Ek, I..5 As Elkies and McMullen point out, these results hold in fact for several test intervals I,..., I m : heorem Elkies and McMullen []. Let I = I I m m be a bounded box. hen there is a probability distribution E, I on Z m 0 such that, for any k = k,..., k m Z m 0 lim λ{α : N I, α = k,..., N I m, α = k m } = Ek, I..6 In other words, heorem states that the point process {N α j α + Z} j N on the torus /N Z converges, as, to a random point process on which is determined by the probabilities Ek, I. As pointed out in [0], this process is the same as for the directions of affine lattice points with irrational shift. It is described in terms of a random variable in the space of affine lattices and is in particular not a Poisson process. he second moments and two-point correlation function, however, coincide with those of a Poisson process with intensity. his is a consequence of the Siegel integral formula, see [] for details in the notation of [], Ek, I = E 0 k, I = E 0,ξ k, I with ξ / Q. Specifically, we have k Ek, I = I + I.7 and k Z 0 k=0 k k Ek, I I = I I + I I..8 he third and higher moments diverge []. It is important to note that Elkies and McMullen considered the full sequence { n mod : n }. emoving the perfect squares n does not have any effect on the limit distribution in heorem, since the set of α for which N I, α is different has vanishing Lebesgue measure as. In the case of the second and higher moments, however, the removal of perfect squares will make a difference and in particular avoid trivial divergence. For I = I I m m and s = s,..., s m C m let M, s := N I, α + s N I m, α + sm dα..9 he main objective of the present paper is to establish the convergence of these mixed moments to the corresponding moments of the limit process, where they exist. he case 4

5 of the second mixed moment implies, by a standard argument, the convergence of the two-point correlation function stated in heorem, cf. Appendix of []. We denote the positive real part of z C by e + z := max{ez, 0}, and set σ := m j= e +s j. heorem 3. Let I = I I m m be a bounded box, and λ a Borel probability measure on with continuous density. Choose s = s,..., s m C m, such that σ < 3. hen, lim M, s = k + s k m + sm Ek, I..0 k Z m 0 Our techniques permit to generalize the above results in two ways: emark. Instead of P we may consider P,c = { n mod : c < n, n / }. for any 0 c <. his setting is already discussed in [, Section 3.5], and the upper bounds obtained in the present paper are sufficient to establish heorem 3 in this case. Note that the limit process is different for each c; it coincides with the limit process E c k, I studied in []. As we point out in [], the second moments of E c k, I are Poisson, and hence heorem holds independently of the choice of c. emark. Although Elkies and McMullen assume that λ is the Lebesgue measure, the equidistribution result that is used to prove heorem, in fact, holds for any Borel probability measure λ on which is absolutely continuous with respect to Lebesgue measure. his follows from atner s theorem by arguments similar to those used by Shah [3]. It is important to note that the limit process E c k, I will be independent of the choice of λ. heorem 3 then follows from the general version of heorem for measures λ with continuous density since in this case, for all upper bounds, it is sufficient to restrict the attention to Lebesgue measure. As discussed in [], the generalization of the above results to λ with continuous density yields the convergence of a more general two-point correlation function, N f = N N i,j= i j f α i, α j, N α i α j + m,. to the Poisson limit. hat is, for all f C 0, lim N f = fα, α, s dα ds..3 Strategy of proof he proof of heorem 3 follows our strategy in []. We define the restricted moments M K, s := N I, α + s N I m, α + sm dα.. max j N I j,α K 5

6 heorem implies that, for any fixed K 0, lim MK, s = k + s k m + sm Ek, I,. k Z m 0 k K where k denotes the maximum norm of k. o prove heorem 3, what remains is to show that lim lim sup M, s M K, s = 0..3 K o establish the latter, we use the inequality M, s M K, s N I,α K N I, α + σ dα.4 where I = j I j and σ = j e +s j. As in the work of Elkies and McMullen, the integral on the right hand side can be interpreted as an integral over a translate of a non-linear horocycle in the space of affine lattices. he main difference is that now the test function is unbounded, and we require an estimate that guarantees there is no escape of mass as long as σ < 3. his means that lim lim sup N I, α + σ dα = 0.5 K N I,α K implies heorem 3. he remainder of this paper is devoted to the proof of.5. 3 Escape of mass in the space of lattices Let G = SL, and Γ = SL, Z. Define the semi-direct product G = G by M, ξm, ξ = MM, ξm + ξ, 3. and let Γ = Γ Z denote the integer points of this group. In the following, we will embed G in G via the homomorphism M M, 0 and identify G with the corresponding subgroup in G. We will refer to the homogeneous space Γ\G as the space of lattices and Γ \G as the space of affine lattices. A natural action of G on is defined by x xm, ξ := xm +ξ. Given an interval I, define the triangle CI = {x, y : 0 < x <, y x I}. 3. and set, for g G and any bounded subset C, N g, C = #C Z g. 3.3 By construction, N, C is a function on the space of affine lattices, Γ \G. Let e Φ t t/ 0 u u = 0 e t/, ñu =, 0, u Note that {Φ t } t and {ñu} u are one-parameter subgroups of G. Note that Γ ñu + = Γ ñu and hence Γ {ñu} u [, Φ t is a closed orbit in Γ \G for every t. 6

7 Lemma 4. Given an interval I, there is 0 > 0 such that for all = e t/ 0, α [, ]: N I, α N ñαφ t, CI + N ñ αφ t, CI 3.5 and, for 3 / α 3 /, N I, α = Proof. he bound 3.5 follows from the more precise estimates in []; cf. also [9, Sect. 4]. he second statement 3.6 follows from the observation that the distance of n to the nearest integer, with n and not a perfect square, is at least n + / + /. A convenient parametrization of M G is given by the the Iwasawa decomposition u v / 0 cos ϕ sin ϕ M = nuavkϕ = 0 0 v / 3.7 sin ϕ cos ϕ where τ = u + iv is in the complex upper half plane H = {u + iv C : v > 0} and ϕ [0, π. A convenient parametrization of g G is then given by H [0, π via the decomposition g =, ξnuavkϕ =: τ, ϕ; ξ. 3.8 In these coordinates, left multiplication on G becomes the group action where for we have: and thus furthermore and and g τ, ϕ; ξ = gτ, ϕ g ; ξg 3.9 a b g =, m c d gτ = u g + iv g = aτ + b cτ + d v g = Imgτ = We define the abelian subgroups { m Γ = 0 Γ = v cτ + d ; 3. ϕ g = ϕ + argcτ + d, 3.3 ξg = dξ cξ, bξ + aξ m. 3.4 } : m Z Γ { } m, 0, m 0 : m, m Z Γ. hese subgroups are the stabilizers of the cusp at of Γ\G and Γ \G, respectively. 7

8 Denote by χ the characteristic function of [, for some, i.e. χ v = 0 if v < and χ v = if v. For a fixed real number β and a continuous function f : of rapid decay at ±, define the function F,β : H by F,β τ; ξ = where f β : G is defined by γ Γ \Γ = γ Γ \Γ f β γg, fξγ + mv / γ v β γ χ v γ 3.5 f β, ξnuavkϕ := fξ v / v β χ v. 3.6 We view F,β τ; ξ = F,β g as a function on Γ \G via the identification 3.8. We show in [, Sect. 3] that there is a choice of a continuous function f 0 with compact support, such that for β = σ, and v with sufficiently large, we have N g, CI σ F,β g = F,β τ; ξ. 3.7 he following proposition establishes under which conditions there is no escape of mass in the equidistribution of translates of non-linear horocycles. In view of Lemma 4 and β 3.7, it implies.5 and thus heorem 3. Use v = / and note that > so the β choice η = is always permitted. Proposition 5. Assume f is continuous and has compact support. Let 0 β < 3. hen lim lim sup F,β ñuav du = v 0 where the range of integration is [, ] for β <, and [, θv η ] [θv η, ] for β and any η [0,, θ 0,. β β he proof of this proposition is organized in three parts: the proof for β <, a key lemma, and finally the proof for β < 3. In the following we assume without loss of generality that f is nonnegative, even, and that frx fx for all r and all x. 4 Proof of Proposition 5 for β < his case is almost identical to the analogous result in []. Since f is rapidly decaying and, we have F,β τ; ξ f F,β τ 4. where hus F,β τ = γ Γ \Γ F,β u + iv; ξ du f v β γ χ v γ. F,β u + iv du

9 he evaluation of the integral on the right hand side is well known from the theory of Eisenstein series. We have c F,β τ = v β v β χ v + c β τ + d + χ v c m β c τ + d c m c= d= gcdc,d= his function is evidently periodic in u = e τ with period one, and its zeroth Fourier coefficient is we denote by ϕ Euler s totient function 0 F,β u + iv du = v β χ v + v β c= ϕc c β t + χ β vc t + dt. 4.4 he first term vanishes for v <, and the second term is bounded from above by v β c= c β t + χ β vc t + with the function K : >0 0 defined by K x = x β t + β χ dt = v / x t + c= K cv / 4.5 dt. 4.6 We have K x max{, x β } max{, x / } and furthermore K x = 0 if x > /. hus K cv / = K xdx, 4.7 lim v 0 v/ c= which evaluates to a constant times β. 5 Key lemma Lemma 6. Let f C be rapidly decreasing and let S = d f 4c + m. 5. D c D d D gcdc,d= hen, for D, > and any ε > 0, we have S where the implied constant depends only on ε and f. D, 5. ε 9

10 Proof. We assume without loss of generality that f is even, non-negative, and of Schwartz class. We prove two statements about S from which the statement of the Lemma will follow. hey are and D c D d D gcdc,d= d 4c +m S D+ε for any ε > S D + D3/ ε for any ε > hen S is bounded by the smaller of these expressions, and it is easy to see that the bound in 5. holds no matter which realizes the minimum. Equation 5.3 is verified by summing over quadratic residues modulo 4c. Note that the conditions d c and gcdc, d = imply d / Z. For coprime D c D and 4c d D and m Z such that d + m 4c, we use rapid decay of f to get d f 4c + m d + m 5.5 A 4c D c D d D A d 4c +m D c D d D \{0} m A 5.6 D A 5.7 for every A >. For a positive integer n, denote by ωn the number of distinct prime factors of n and by τn the number of divisors of n. For coprime D c D and d D and m Z such that d + m <, we have 4c denote by the distance to the nearest integer since D c D d D d < gcdc,d= 4c +m f d 4c + m D c D d D gcdc,d= f D c D j c j is a square mod 4c d 4c ω4c f 5.8 j, 5.9 4c #{d mod 4c : d j mod 4c, gcdc, d = } ω4c 5.0 for every j. Now ω4c is the number of squarefree divisors of 4c and is therefore at most τ4c. Combined with rapid decay we use ft, this yields t D c D j 4D τ4c c j. 5. 0

11 Finally the fact that for every ε > 0, τn n ε gives which is D+ε D+ε log D 5. for every ε > 0. It suffices to note that to verify 5.3. Inequality 5.4 is obtained as follows. he Poisson summation formula yields S c 0 D + nd c n e c n 0 D c D d 4c Here c n = ˆf n 5.4 where ˆf is the Fourier transform of f, which is also of Schwartz class, and ez = e πiz is the usual shorthand. he second term in 5.3 is bounded by n 0 D c D d 4c nd c n e 4c 8D r= n 0 n 0 4D/r c 8D/r gcdn,4c= c nrr r 8D 4D/r c 8D/r d mod 4c nd e 5.5 4c c nr r 8c. 5.6 he last inequality follows from the well known evaluation the classical Gauss sum with gcdn, 4c = nd 4c 4c, e = + i ε n 5.7 4c n d mod 4c where 4c n is the Jacobi symbol and εn = or i if n = or 3 mod 4, respectively. When nr < we use the fact that the Fourier transform of f is bounded: herefore, 5.6 D3/ n 0 r 8D nr < c nr. 5.8 r = D3/ r 8D r n </r D 3/ 5.9 When nr >, we use the fact that the Fourier transform of f decays faster than any polynomial since f is smooth: c nr n A r A A. 5.0

12 We take A >. hen we have 5.6 D 3/ A D 3/ A his proves 5.4 and the Lemma. nr > n 0 r 8D k= n A r A+/ D3/ A nr> n,r nr A 5. k A+ε D 3/ ε. 5. We have 6 Proof of Proposition 5 for β where ξ = u/, u /4. For this choice we have ñuav = u + iv, 0; ξ 6. F,β τ; ξ = f m + u /4 v/ v β τ τ χ β + f cu v / /4 + du/ + m cτ + d c,d Z gcdc,d= c>0,d 0 v + 6. τ v β cτ + d χ β v cτ + d. 6.3 he integral of the first term tends to zero as v 0. We write = v/ τ = v/ u +v. Indeed, for m 0 we have fm + u /4 m A from rapid decay, so that J 6.du A+β du v / v A+β du v A β, 6.4 where J = [, θv η ] [θv η, ]. If A > β, then this contribution is negligible as v 0. For m = 0, we have f so that the contribution of this term is, assuming β >, J β du θv η u β v β/ du = θ β β vβ/+η βη since η < β ; similarly for β =. β It remains to analyze the contribution of 6.3. Notice that the this term is nonzero only when d/c is in the range of integration for u, which is contained in the interval [, ]. herefore we restrict the summation to 0 < d c. Now we perform the substitution t = u + d/c v to zoom in on each rational point and extend the range of integration

13 to all of. his gives t= c f 4 d + tv c + d d + tv + m c c vt + v c vt + χ β and we need to bound f d + m + Octv 4c c vt + c= t 0< d c c,d= Now we decompose the region c vt + v dt c vt + χ β c vt + into dyadic regions j c vt + < j+ dt 6.6 c vt for j log. We can thus bound 6.7 by j log c 0< d c c,d= f d 4c + m + Octv c vt + v c vt + χ β [ j, j+ dt c vt + j log c 0< d c c,d= f j d 4c + m + Octv v c vt + χ β [ j, j+ dt c vt + v βj j log f j+ vt c j + vt 0< d j + vt + c,d= j d 4c + m + Octv dt. 6.8 It remains to remove the error term Octv from the argument of f to apply Lemma 6. We have that j ctv v / for every j. Define f x = max fx + y. y 3

14 hen, f x fx + Ov / for v small enough, and we can bound 6.8 by a similar expression with f j d + m in place of f j. d + m + Octv o bound this 4c 4c we apply Lemma 6 with D 6.8 v as by our choice of ε. vt j+, = + j, and ε = 3 β > 0. hen we have j log j log βj D dt 6.9 ε t jβ 3+ε = jβ 3/ j log eferences [] Daniel El-Baz, Jens Marklof, and Ilya Vinogradov. he distribution of directions in an affine lattice: two-point correlations and mixed moments. Int Math es Notices 03 doi: 0.093/imrn/rnt58, 03. [] Noam D. Elkies and Curtis. McMullen. Gaps in n mod and ergodic theory. Duke Math. J., 3:95 39, 004. [3] A Eskin, G Margulis, and S Mozes. Quadratic forms of signature, and eigenvalue spacings on rectangular -tori. Ann. of Math, :6, 005. [4] D.. Heath-Brown. Pair correlation for fractional parts of αn. Math. Proc. Cambridge Philos. Soc., 483: , 00. [5] Gregory Margulis and Amir Mohammadi. Quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms. Duke Math. J., 58: 60, 0. [6] J. Marklof and A. Strömbergsson. Equidistribution of Kronecker sequences along closed horocycles. Geom. Funct. Anal., 36:39 80, 003. [7] Jens Marklof. Pair correlation densities of inhomogeneous quadratic forms. II. Duke Math. J., 53: , 00. [8] Jens Marklof. Pair correlation densities of inhomogeneous quadratic forms. Ann. of Math., 58:49 47, 003. [9] Jens Marklof. Distribution modulo one and atner s theorem. In Equidistribution in number theory, an introduction, volume 37 of NAO Sci. Ser. II Math. Phys. Chem., pages Springer, Dordrecht, 007. [0] Jens Marklof and Andreas Strömbergsson. he distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. Ann. of Math., 73: , 00. [] Zeév udnick and Peter Sarnak. he pair correlation function of fractional parts of polynomials. Comm. Math. Phys., 94:6 70, 998. [] Peter Sarnak. Values at integers of binary quadratic forms. In Harmonic analysis and number theory Montreal, PQ, 996, volume of CMS Conf. Proc., pages Amer. Math. Soc., Providence, I,

15 [3] Nimish A Shah. Limit distributions of expanding translates of certain orbits on homogeneous spaces. Indian Academy of Sciences. Proceedings. Mathematical Sciences, 06:05 5, 996. [4] Ya. G. Sinai. Statistics of gaps in the sequence { n}. In Dynamical systems and group actions, volume 567 of Contemp. Math., pages Amer. Math. Soc., Providence, I, 0. [5] Jeffrey M. Vanderkam. Pair correlation of four-dimensional flat tori. Duke Math. J., 97:43 438, 999. [6] Jeffrey M. Vanderkam. Values at integers of homogeneous polynomials. Duke Math. J., 97:379 4, 999. [7] Jeffrey M. VanderKam. Correlations of eigenvalues on multi-dimensional flat tori. Communications in Mathematical Physics, 0:03 3, 000. Daniel El-Baz, School of Mathematics, University of Bristol, Bristol BS8 W, U.K. daniel.el-baz@brisol.ac.uk Jens Marklof, School of Mathematics, University of Bristol, Bristol BS8 W, U.K. j.marklof@bristol.ac.uk Ilya Vinogradov, School of Mathematics, University of Bristol, Bristol BS8 W, U.K. ilya.vinogradov@bristol.ac.uk 5