L p and Weak L p estimates for the number of integer points in translated domains

Transcription

1 Under consideration for publication in Math Proc Camb Phil Soc 1 L p and Weak L p estimates for the number of integer points in translated domains By LUCA BRANDOLINI Dipartimento di Ingegneria Gestionale, dell Informazione e della Produzione, Università di Bergamo, Viale Marconi 5, Dalmine BG, Italy lucabrandolini@unibgit LEONARDO COLZANI Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Cozzi 53, Milano, Italy leonardocolzani@unimibit GIACOMO GIGANTE Dipartimento di Ingegneria Gestionale, dell Informazione e della Produzione, Università di Bergamo, Viale Marconi 5, Dalmine BG, Italy giacomogigante@unibgit GIANCARLO TRAVAGLINI (corresponding author Dipartimento di Statistica e Metodi Quantitativi, Università di Milano-Bicocca, Via Bicocca degli Arcimboldi 8, Milano, Italy giancarlotravaglini@unimibit (Received Abstract Revisiting and extending a recent result of MHuxley, we estimate the L p ( T d and Weak L p ( T d norms of the discrepancy between the volume and the number of integer points in translated domains In this paper we estimate different norms of the discrepancy between the volume and the number of integer points in dilated and translated copies RΩ x of a bounded convex domain Ω R d having positive measure The above number of integer points is a periodic function of the translation variable x, with Fourier expansion χ RΩ x (k = χ RΩ y (k exp( πinydy exp(πinx k Z d n Z d T d k Z d 010 Mathematics subect classification 11H06, 5C07 Key words and phrases Lattice points, Discrepancy

2 Brandolini, Colzani, Gigante, Travaglini = χ RΩ (y exp( πinydy exp(πinx ( R n Z d d = n Z d R d χ Ω (Rn exp(πinx These equalities are in the L sense It follows that the discrepancy function D (RΩ x = k Z d χ RΩ x (k R d Ω has Fourier expansion n Z d \{0} R d χ Ω (Rn exp(πinx If Ω is a bounded convex domain in R d with smooth boundary having positive Gaussian curvature then χ Ω (ξ C ξ (d+1/ See [16, Chapter 8] Kendall [11] observed that the Fourier expansion of the discrepancy and the above estimate for the Fourier transform of a convex domain give } 1/ D (RΩ x dx CR (d 1/ T d Using a smoothing argument and the Poisson summation formula, Herz [8] and Hlawka [9] (see also [18] proved that sup { D (RΩ x } CR d(d 1/(d+1 x T d Interpolating the above two upper bounds between L and L gives a poor estimate Indeed when d = interpolation gives } 1/p D (RΩ x p dx CR (p 1/(3p, T while MHuxley [10] has recently showed a more interesting estimate: If Ω is a planar convex body having boundary with continuous and positive curvature then } 1/4 D (RΩ x 4 dx CR 1/ log 1/4 (R T That is, the upper estimate for the L discrepancy extends, up to a logarithm, to L 4 Huxley s proof seems to be tailored for the planar case and for the exponent p = 4, where one can apply Parseval equality to the square of the discrepancy function Huxley also asked for an analog of his result for d > Here we will give a possible answer and our approach will be to obtain L p results through Weak L p techniques We recall that the spaces L p (X, µ and Weak L p (X, µ, 0 < p < +, are defined by the quasi norms 1/p f L p (X,µ = f (x p dµ (x}, X f Weak L p (X,µ = sup {t p µ {x X, f (x > t}} 1/p t>0

3 Integer points in translated domains 3 The space Weak L p (X, µ is the case q = + of the Lorentz spaces L p,q (X, µ (see eg [1, Chapter 1, 3] or [17, Chapter 5, 3] Finally, the space L (X, µ is defined by the norm f L (X,µ = inf {t > 0 : µ {x X : f (x > t} = 0} In what follows (X, µ will be the torus T d or the integers Z d with the respective translation invariant measures If X has finite measure and p < s, then both L p (X, µ and L s (X, µ are intermediate between L (X, µ and Weak L p (X, µ: L (X, µ L s (X, µ L p (X, µ Weak L p (X, µ The following is a sharp quantitative counterpart of these inclusions Lemma 1 (1 If X has finite measure, then f p L p (X,µ f p Weak L p (X,µ ( If p < s < +, then Proof (1 Observe that f s L s (X,µ { 1 + log ( p } µ (X f L (X,µ f p Weak L p (X,µ s s p f p Weak L p (X,µ f s p L (X,µ µ {x X, f (x > t} µ {X}, µ {x X, f (x > t} f p Weak L p (X,µ t p, µ {x X, f (x > t} = 0 if t f L (X,µ Hence, + f p L p (X,µ = pt p 1 µ {x X, f (x > t} dt 0 pµ (X µ(x 1/p f Weak L p (X,µ 0 f L (X,µ t p 1 dt + p f p dt Weak L p (X,µ µ(x 1/p f Weak L p (X,µ t { ( p } µ (X f = f p L Weak L p (X,µ 1 + log (X,µ f p Weak L p (X,µ ( As before, + f s L s (X,µ = st s 1 µ {x X, f (x > t} dt 0 f L (X,µ s f p Weak L p (X,µ 0 t s 1 p dt = s s p f p Weak L p (X,µ f s p L (X,µ

4 4 Brandolini, Colzani, Gigante, Travaglini The first inequality in the above lemma is sharp Indeed, if the measure is not atomic, one can always choose a function with distribution function that turns the above inequalities into equalities Also the second inequality is sharp if the measure is not atomic and infinite, but it can be slightly improved when the measure is finite Our first result is a simple application of the Hausdorff-Young inequality Theorem Let Ω be a bounded open set in R d (1 If p < + and 1/p + 1/q = 1, then D (RΩ x L p (T d { χω Rd Lq (Rn} (Z d ( If < p < + and 1/p + 1/q = 1, then D (RΩ x Weak Lp (T d CRd { χω (Rn} Weak Lq (Z d (3 If < p < + and 1/p + 1/q = 1, then D (RΩ x L p (T d CRd log 1/p Weak L ( + R { χ Ω (Rn} q (Z d (4 If 1 p +, then D (RΩ x Lp (T d { sup Rd χ Ω (Rn } Proof Point (1 readily follows from the Fourier expansion of the discrepancy and the Hausdorff-Young inequality: If p + and 1/p + 1/q = 1, then f L p (T d f L q (Z d The case (p, q = (, is Parseval s identity The case (p, q = (+, 1 is immediate The intermediate cases follow by the Riesz-Thorin interpolation theorem See [1, Theorem 111] or [17, Chapter V, 1] Similarly, point ( follows from the Hausdorff-Young inequality for Lorentz spaces: If < p < + and if 1/p + 1/q = 1, then f Weak Lp (T d C f Weak L q (Z d The proof of this inequality is by real interpolation between the extreme cases L L and L 1 L See the general Marcinkiewicz interpolation theorem [1, Theorem 53] or [17, Chapter V, 3] Point (3 follows from point (, Lemma 1, and the trivial estimate D (RΩ x CR d Finally, a Fourier coefficient is dominated by the norm of the function, and point (4 follows The above theorem is quite abstract In order to obtain explicit results, one has to estimate the norms of the sequences { χ Ω (Rn} The interest in case (3 is when the L ( q Z d norm is infinite and the Weak L ( q Z d norm is finite In order to introduce the next result, we recall that the modulus of continuity of a characteristic function shows that such a function does not belong to a Sobolev class W ( α, R d whenever α 1/ See [15, Chapter 5, 5] Moreover, in [1, Corollary ] it is proved that for every set Ω with finite positive measure, without any regularity assumption, there exists a constant C such that χ Ω (ξ dξ CR 1 ξ >R

5 Integer points in translated domains 5 It follows that a uniform inequality of the kind χ Ω (ξ C ξ β cannot hold with β > (d + 1 / On the other hand, this estimate holds with β = (d + 1 / if Ω is a bounded convex domain with smooth boundary with non-vanishing Gaussian curvature See [16] See also [6] for possible generalizations to convex bodies with smooth boundary containing isolated points with vanishing Gaussian curvature This motivates the following Corollary 3 Assume that Ω is a bounded convex domain such that (1 If p < d/ (d 1 and R >, then ( If p d/ (d 1 and R >, then (3 If p = d/ (d 1 and R >, then χ Ω (ξ C ξ (d+1/ D (RΩ x Lp (T d CR(d 1/ D (RΩ x Weak L p (T d CR(d 1/ D (RΩ x L p (T d CR(d 1/ log (d 1/(d (R (4 If p > d/ (d 1 and R >, then D (RΩ x Lp (T d CRd(pd p d+1/p(d+1 Proof Points { (1, ( and (3 follow from Theorem, and the observation that the sequence n α} is in L ( q Z d if and only if qα > d, and it is in Weak L ( q Z d if and only if qα d Point (4 follows from point ( with p = d/(d 1, the pointwise estimate D (RΩ x CR d(d 1/(d+1 proved in [8] and [9], and ( in Lemma 1 The estimates in the above Corollary for p < d/ (d 1 are essentially sharp In order to show this, we first recall the following result on the Fourier transform of the characteristic function of a convex set Theorem 4 Let Ω R d be a convex body with smooth boundary having everywhere positive Gaussian curvature For every ξ R d \ {0} let σ (ξ be the unique point on the boundary Ω with outward unit normal ξ/ ξ Also let K (σ (ξ be the Gaussian curvature of Ω at σ (ξ Then, as ξ +, the Fourier transform of χ Ω (x has the asymptotic expansion χ Ω (ξ = 1 + O d+1 πi ξ ( ξ d+3 Proof See eg [7], [8], or [9] [K 1 (σ (ξ e πi(ξ σ(ξ d 1 8 K 1 (σ ( ξ e πi(ξ σ( ξ+ d 1 8 ] The following result partially complements Corollary 3 Theorem 5 Let Ω R d be a convex body with smooth boundary having everywhere positive Gaussian curvature

6 6 Brandolini, Colzani, Gigante, Travaglini (1 If Ω is not symmetric about a point, or if the dimension d 1 (mod 4, then for every p 1 there exists C > 0 such that for every R >, D (RΩ x Lp (T d CR d 1 ( If Ω is symmetric about a point and if d 1 (mod 4 then } lim sup {R d 1 D (RΩ x L p (T d > 0 if p 1, } lim inf {R d 1 D (RΩ x Lp (T d = 0 if p < d d 1 More precisely, if p < d/ (d 1 there exist C > 0, and a sequence R +, such that D (R Ω x L p (T d d 1 CR ( d 1 log (R d 1 p log (log (R Proof In order to prove point (1 observe that, by Theorem, { D (RΩ x L p (T d sup Rd χ Ω (Rn } Moreover, by Theorem 4, for n 0, R d χ Ω (Rn = 1 πi R d 1 n d+1 + O [K 1 (σ (n e πi(rn σ(n d 1 8 K 1 (σ ( n e πi(rn σ( n+ d 1 8 ] (R d 3 n d+3 If Ω is not { symmetric, then } also K (σ (u is not symmetric (see [, 14, p 133] Since n the set n : n Zd \ {0} is dense in the unit sphere, by continuity there exists m Z d such that K (σ (m K (σ ( m Then, for this m and R large enough, Rd χ Ω (Rm 1 ( π R d 1 m d+1 K 1 (σ (m K 1 (σ ( m + O R d 3 m d+3 CR d 1 Assume now that Ω is symmetric, and translate the center of symmetry to the origin, so that for every ξ we have σ ( ξ = σ (ξ and K (σ (ξ = K (σ ( ξ Choose n 0 and observe that n (σ (n σ ( n = n σ (n 0 Indeed, n σ (n = 0 would imply that the center belongs to the hyperplane tangent to Ω at σ (n, hence Ω should have measure 0 We have R d χ Ω (Rn ( CR d 1 n d+1 K 1 e (σ (n πi(rn σ(n d O R d 3 n d+3 Let x denote the distance of a real number x from the integers If d 1 Rn σ (n 4 > 1 10,

7 Integer points in translated domains 7 d 1 πi(rn σ(n then e 4 1 > c and we have Assume now that Then Rd χ Ω (Rn cr d 1 d 1 Rn σ (n d 1 4Rn σ (n 1 5 Since d 1 (mod 4, we have d 1 4Rn σ (n Applying the previous argument with n in place of n provides the estimate Rd χ Ω (Rn d 1 cr In order to prove point (, assume that Ω is symmetric and d 1 (mod 4 From the asymptotic estimate of χ Ω (ξ we obtain R d χ Ω (Rn = 1 ( π R d 1 n d+1 K 1 (σ (n sin (πrn σ (n + O R d 3 n d+3 Since n σ (n 0, lim sup lim sup } {R d 1 D (RΩ x L p (T d {R d 1 Rd χ Ω (Rn } = 1 d+1 n K 1 (σ (n > 0 π The last part of the proof relies on the ideas of Parnovski and Sobolev in [13] We need a variant of Dirichlet s theorem on simultaneous diophantine approximation (see [13] Let α 1,, α m be real numbers, then for every positive integer there exist integers s 1,, s m and r such that Let β = r m+1 and rα k s k < 1 for every k = 1,, m p d pd+p, and {α k } m k=1 = {n σ (n} n β Then there exist integers s n and R such that It follows that R cβd +1 Cβd and R n σ (n s n 1 sin (πr n σ (n = sin (π (R n σ (n s n π 1 By the Hausdorff-Young inequality in Theorem, for p and 1/p + 1/q = 1, we have } 1/p D (R Ω x p dx Rd χ T Ω (Rn 1/q q d 0 n Z d

8 8 Brandolini, Colzani, Gigante, Travaglini If p < d d 1 then the above estimates of χ Ω (R n yield R d χ Ω (R n q CR (d 1q/ n (d+1q/ sin (πr n σ (n q + ( O CR (d 1q/ q n (d+1q/ + 0< n ( β CR (d 1q/ q + β(d (d+1q/ ( + O Since β = n > β n (d+1q/ R (d 3q/ q q(d+1 d and R, we obtain R d χ Ω (R n 1/q q cr (d 1/ 1 m 0 Finally, letting + we obtain } lim inf {R d 1 D (RΩ x Lp (T d lim inf d 1 R R (d 3q/ ( + O n (d+3q/ R (d 3q/ R d χ Ω (Rn 1/q q = 0 More precisely, if ϕ (t = t βd log (t then one can prove that, for large s, ( 1/βd βds ϕ 1 (s log (s This implies that if R Cβd then Therefore m 0 1 C ( 1/βd log (R log (log (R R d χ Ω (R n 1/q q C R (d 1/ ( 1/βd log (R log (log (R As we said, χ Ω (ξ C ξ (d+1/ whenever Ω has smooth boundary with positive Gaussian curvature However, for domains in the plane this smoothness assumption can be relaxed Consider a convex body Ω which can roll unimpeded inside a disc This means that for any point x on the boundary there is a translated copy of Ω contained in that touches in x Theorem 6 If a planar convex set Ω can roll unimpeded inside a disc, then χ Ω (ξ C ξ 3/ Proof In [14] (see also [5, 18] it is proved that if { λ (δ, ϑ, Ω = x Ω : δ + x ϑ = sup {y ϑ}} y Ω

9 Integer points in translated domains 9 ϑ Ω δ (this is the length of the chord perpendicular to the outward direction ϑ and at a small distance δ from the boundary Ω, then χ Ω (ρϑ diameter(ω ρ ( λ ( (ρ 1, ϑ, Ω + λ ( (ρ 1, ϑ, Ω If Ω can roll unimpeded inside a disc, then λ (δ, ϑ, Ω λ (δ, ϑ, This implies that the Fourier transform of Ω is dominated by the chords of a disc, and therefore χ Ω (ξ C ξ 3/ A curve can roll unimpeded inside another curve if and only if the largest radius of curvature of the first is smaller than the smallest radius of curvature of the second No smoothness of these curves is required, the rolling curve may also have corners See [, Chapter 17] and the references therein In particular, the above results give an alternative proof of the result in [10] Corollary 7 Let Ω be a planar convex set that can roll unimpeded inside a disc For any R > we have } 1/4 D (RΩ x 4 dx CR 1/ log 1/4 (R T We conclude with a remark A spherical shell { x R d : R x < R + 1 } { contains } approximately R d 1 integer points which lie on the spherical surfaces x = n with n integer between R and (R + 1 Since these integers are at most R+1, one of these surfaces contains at least cr d integer points This implies that, if Ω = { x R d : x 1 }, then D(RΩ x L (T d crd for a diverging sequence of R On the other hand we proved that D(RΩ x Lp (T d cr(d 1/ for p < d/(d 1 Since for d > 3 this L p estimate is smaller than the L one, there exists a critical index p for which D(RΩ x Lp (T d cr(d 1/ starts failing We do not know if this critical index is d/(d 1 REFERENCES [1] JBergh, JLöfström, Interpolation spaces, Spinger Verlag, 1976 [] TBonnesen, WFenchel, Theory of Convex Bodies, BCS Associates, Moscow, Idaho, USA, 1987

10 10 Brandolini, Colzani, Gigante, Travaglini [3] L Brandolini, G Gigante, G Travaglini, Irregularities of distribution and average decay of Fourier transforms, in W Chen, A Srivastav, G Travaglini (eds, A Panorama of Discrepancy Theory, Lecture Notes in Mathematics 107, Springer International Publishing Switzerland, 014 [4] L Brandolini, S Hofmann, A Iosevich, Sharp rate of average decay of the Fourier transform of a bounded set, Geometric and Functional Analysis 13 (003, [5] L Brandolini, M Rigoli, G Travaglini, Average decay of Fourier transforms and geometry of convex sets, Revista Matemática Iberoamericana 14 (1998, [6] J Bruna, A Nagel, S Wainger, Convex Hypersurfaces and Fourier Transforms, Annals of Mathematics 17 (1988, [7] I M Gelfand, M I Graev, N Y Vilenkin, Generalized functions, Vol 5: Integral geometry and problems of representation theory, Academic Press, New York, 1966 [8] C Herz, On the number of lattice points in a convex set, American Journal of Mathematics 84 (196, [9] E Hlawka, Uber Integrale auf convexen Körpen, I, II, Monatshefte für Mathematik 54 (1950, 1-36, [10] M N Huxley, A fourth power discrepancy mean, Monatshefte für Mathematik 73 (014, [11] D G Kendall, On the number of lattice points inside a random oval, Quarterly Journal of Mathematics Oxford 19 (1948, 1-6 [1] M N Kolountzakis, T Wolff, On the Steinhaus tiling problem, Mathematika, 46 (1999, [13] L Parnovski, A V Sobolev, On the Bethe-Sommerfeld conecture for the polyharmonic operator, Duke Math J 107 (001, [14] A N Podkorytov, On the asymptotics of the Fourier transform on a convex curve, Vestnik Leningrad University Mathematics 4 (1991, [15] E M Stein, Singular Integrals and Differentiabilty Properties of Functions, Princeton University Press, 1970 [16] E M Stein, Harmonic Analysis, Princeton University Press, 1993 [17] E M Stein, G Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, 1971 [18] G Travaglini, Number theory, Fourier analysis and geometric discrepancy, Cambridge University Press, 014