Non-isotropic distance measures for lattice-generated sets

Publ. Mat. 49 (2005), 225–247 NON-ISOTROPIC DISTANCE MEASURES FOR LATTICE-GENERATED SETS Alexander Iosevich and Misha Rudnev Abstract We study distance measures for lattice-generated sets in Rd,d≥3, with respect to non-isotropic distances |·|K, induced by smooth symmetric convex bodies K. An effective Fourier-analytic approach is developed to get sharp upper bounds for the second moment of the weighted distance measure. The implications of these estimates are discussed in the context of the general Erd¨os-Falconer distance problem. 1. Introduction 1.1. In this paper we study distance sets, corresponding to the integer lattice Zd,d≥3, with respect to non-isotropic distances, defined in terms of the Minkowski functional of some well-curved central-symmetric convex body K⊂Rd, with a smooth boundary ∂K. Namely we assume that the Gaussian curvature on the boundary ∂K is bounded from above and below by some fixed pair of positive constants. The boundary itself should be Cr, for a large enough r, and we do not discuss how small r can possibly be. Suppose the volume of Kequals the volume of the Euclidean unit ball, which is denoted as Bthroughout the paper. Let K denote the class of such convex bodies. For K∈ K, let |·|Kbe the Minkowski functional of K, or the K-norm. Let |·|K∗be the dual norm to |·|K, defined as (1) |x|K∗= sup y∈K|x·y|, K∗={x∈Rd:|x|K∗≤1}. The dual K∗also belongs to the class K. For a Borel set S⊂Rd, define its K-distance set as (2) ∆K(S) = {|a−b|K:a, b ∈S}. 2000 Mathematics Subject Classification. 42B. Key words. Lattice point distribution, mean square estimates, distance measures, homogeneous sets. 226 A. Iosevich, M. Rudnev Let ∆(S)≡∆B(S) be the distance set of Swith respect to the Euclidean metric k·k. An infinite discrete set A⊂Rdis called homogeneous if all its elements are separated by some c > 0, while any cube of side length C > c contains at least one element of A. Let q1 be a large real, consider a homothety qK of K, with respect to the center of K. For a homogeneous discrete set A, let Aq=A∩qK be a truncation of A(which strictly speaking depends on K). In the special case of A=Zdand K=B, it is well known that (3) #∆(Aq)≈   q2, d ≥3, q2 √log q, d = 2. To fix the notation, # denotes the cardinality of a finite set, |·|stands for the Lebesgue measure of a Borel measurable set. The symbols O or .absorb constants depending only on K(and hence d). Also we write a&b, or equivalently a= Ω(b), if b.aand a≈bif both a.b and a&b. The symbol ∼will indicate proportionality, up to some constant c(K). The main goal in this paper is to develop a technically transparent Fourier-analysis based approach which would extend estimate (3) to the case of K-distances. We are able to do so in d≥4; if d= 3 we are off by a logarithmic factor. The result follows by estimating the second moment of the corresponding distance measure; it is stated in Theorem 2 further in the paper, after all the notations have been developed. Theorem 1. If A=Zd, the d-dimensional integer lattice, one has (4) #∆K(Aq)&(q2, d ≥4, q2log−2q, d = 3. 1.2. Theorem 1 can be given interpretation in terms of the borderline dimension d 2in the Falconer distance problem. The Falconer distance problem states that if the Hausdorff dimension of a Borel set S⊂Rd, d≥2 is greater than d 2, then the Lebesgue measure of the distance set |∆(S)|>0. (See [5], [11], [2], [16], [17], [3], and the references contained therein for the description of this open for every d≥2 problem and progress over the years. The best known results are due to Wolff in R2and Erdo˜gan in Rd, who vindicate the conjecture if the dimension of Sexceeds d(d+2) 2(d+1) rather than d 2.) The discrete analogue of the Falconer distance problem is the Erd¨os distance conjecture (see e.g. [13] and the Non-Isotropic Lattice Distance Measures 227 references contained therein for thorough discussion and the state-of-theart) restricted to homogeneous sets, which states (5) #∆(Aq)≥Cεq2−ε. Falconer [5] showed that the borderline dimension d 2cannot be improved, due to the following construction. Fix a rapidly growing sequence of positive integers {qi}i≥1,with q1= 2 and qi+1 > qi i. Let A=Zd and Sibe the union of Euclidean balls of radius q−d s i, for some 0 < s < d, centered at the points of 1 qiAqi. Let us call SF=∩iSithe Falconer set. Then (see e.g. [4]) the Hausdorff dimension dimHSF=s. On the other hand, the Lebesgue measure (6) |∆(Si)| ≈ q−d s i·#∆(Aqi). It follows from (3) that |∆(SF)|= 0 if s < d 2. More precisely, |∆(SF)|>0, provided that s≥d 2for d≥3 and s > d 2for d= 2. An immediate consequence of Theorem 1 is that the same conclusion can be drawn in the case d≥4 for the Falconer set SFwith respect to K-distances, K∈ K, irrespective of the pair of constants that bound the curvature. Consequently, as the basis for the Falconer construction one can use any d-dimensional lattice, in which case Theorem 1 will be valid as well. Corollary 1.1. Let SFbe the Falconer set with dimHSF=s. Suppose, s≥d 2, and d≥4. Then |∆K(SF)|>0, for any central-symmetric K, which is smooth and has bounded and everywhere non-zero curvature on ∂K. Remark. The assumption of central symmetry is not essential: one can always consider Zd +only. We do not study to what extent the curvature assumption can be weakened so that Kmay still qualify as “well-curved”, which for our purposes is determined by the validity of Lemma 2.1 further in the paper. As a separate issue, our result shows that if one fixes the Euclidean metric k· k, the scope of Falconer’s construction can be extended to a class Aof homogeneous sets (7) A=|a|K kaka:a∈Zd, K ∈ K. The difficult part in Theorem 1 is the endpoint issue, namely the precision of the exponents in estimates (4), which transcribes into the inequality s≥d 2, inclusive of the endpoint, in Corollary 1.1. Otherwise, 228 A. Iosevich, M. Rudnev that is if an extra qεwere allowed in the right-hand side of (4), the proof can be made somewhat shorter, using the techniques by M¨uller [12] and Iosevich et al. [8] developed to study the quantity (8) E(t) = #{tK ∩Zd}−tdVol K. 1.3. It appears to be important to understand how much the analytic methods for geometry of numbers, i.e. in the present context the case A=Zd, can apply to study the Erd¨os-Falconer problem, dealing with general homogeneous sets A. The motivation for doing so it comes from a hypothesis closely related to (5) that lattice sets yield local minima for second moments of the corresponding distance measures, with respect to variations of the sets. Hence our approach is developed on the basis of the general distance measure formalism, set up by Mattila [11]. The distance measure ν(t), relative to the set Aqcounts the number of points of Aqin 1 q-thin K-annuli of radius t, centered at points of Aq, average with respect to the position of the center. (In the case of a lattice it suffices to fix the center at the origin.) The L1-norm kνk1is approximately the number of points of Ain qK. The main task is to estimate the square of L2-norm, or the second moment kνk2 2. This is the content of the forthcoming Theorem 2, after all the definitions have been made. Beyond this non-technical introduction, we will be using various weighted measures ν, which will carry extra identification. The distance measure formalism however has nothing to do with the lattice structure, in the sense that any finite compactly supported Borel measure µin Rdgenerates a well-defined distance measure νin R+. To this effect, Mattila [11] proved a general theorem for the Euclidean distance, which generalizes to K-distances (see [1]) as follows. The second moment of the K-distance measure ν, generated by µis finite in case what we call the Mattila integral (9) M(µ) = Z∞ 1Z|ˆµ(tx)|2dωK∗(x)2 td−1dt < ∞. Above, ωK∗is the Lebesgue measure on ∂K∗. Hence, if M(µ)<∞, the Lebesgue measure of the support of νis positive. As for the Falconer construction (see 1.2), for the natural (i.e. induced by the Lebesgue measure in Rd) measure on the set SF, in the case dimHSF<d 2the integral (9) diverges. In essence, our proof of Theorem 2 consists in analyzing and estimating the Mattila integral for a natural measure µon the set Si=Sqi,in the Falconer construction, when SF=∩Sihas Hausdorff dimension s=d 2, after fixing qi=qand appropriate scaling. Non-Isotropic Lattice Distance Measures 229 Looking back at the construction of the set SF, it is clear that instead of the lattice Zd,one can use any homogeneous set Aas a basis for the construction. Unfortunately, our proof of Theorem 2 does not extend beyond the very special case when Ais a lattice. The reason is that in the proof we use a smooth approximation E(t) of the discrepancy E(t), see (8), as the auxiliary quantity. In the lattice case, E(t) admits a well known analytic representation via the Poisson summation formula, whereupon E2(t) looks very similar to the integrand in the Mattila integral. It follows that L2estimates for the distance measure νcan be obtained in terms of L2-estimates for E. Whether such an approach has a prototype in the general homogeneous set context is not at all clear. 1.4. To prove our main result we develop an asymptotic method, which enables one first to dominate the L2-estimate for Eby a weighted L2estimate for ν, see (46) below. Then the latter estimate can in turn be dominated by another L2-estimate for E, see e.g. (62) below. Theorem 2 follows. L2-estimates for the quantity Ewere obtained in the works [12] and [8] (see also [9] for further developments) where basically the same trick was used. However, the asymptotic techniques of those papers did not yield a clear cut relation like (46) between the L2estimates for the quantities νand E, due to plethora of cut-off functions, truncations, etc. used. These are the technical difficulties one encounters in the effort to attain the endpoint result claimed in Theorem 1. We identify the estimate (46) as the key display of the technical advantage of our approach, which also yields the mean square estimates for Eand Eas a by-product. In addition, throughout the proof a number of integral representations for the distance measure νand related quantities are obtained, which can be interesting in their own right. The approach rests on the use of the Hankel rather than Fourier transform for distance measures, defined on R+, which enables to make the analysis fairly transparent. Due to the fact that it is only a weighted estimate for the second moment of the distance measure νthat gets majorated by the second moment of E, our approach results in tight (modulo the logarithmic factor in d= 3) estimates for the second moment of νin d≥3,yet for d= 2 it does not do better than yield a trivial estimate. In d= 2, the case of a general Kis an open problem. The main body of the paper is organized as follows. In Section 2 we set up the distance measure formalism in the context of a general homogeneous set A. In the special case A=Zd, we formulate Theorem 2 and show how it implies Theorem 1. As an example of how the formalism applies to a general A, we briefly discuss the Euclidean distance case 230 A. Iosevich, M. Rudnev and write out the integral expression for the second moment of the distance measure. We further move on to the case of K-distances and prove the general Mattila integral identity for the second moment in Proposition 2.2. The proposition contributes little to the special case A=Zd. However, it establishes the proof template which is further used in Section 3 to prove the crucial Lemma 3.1. Section 3 however is already fully dedicated to the case A=Zdand from its outset takes advantage of the Poisson summation formula. Comparison of the yield of the Poisson summation formula for the distance measure with the general formula in Proposition 2.2 yields as a by-product Theorem 3 on duality. However, the main result of Section 3 is Lemma 3.1. In Section 4 this lemma is used to prove Theorem 2. 2. Distance measure Let φbe a non-negative radial (radial henceforth means radial with respect to the Euclidean metric) Schwartz class function, such that Rφ(x) = 1, φ(x) = 1 inside the ball of some radius and vanishes outside the ball of twice the radius. Let qbe a large number, denote φq(x) = qdφ(qx) and A⊂Rda homogeneous set, Aq=A∩qK. Let also Zd q=Zd∩qK for the special case A=Zd. Without loss of generality (only to discount additional trivial estimates) suppose Aqcontains no point in some c-neighborhood of the origin. For a function f∈L1(Rd)∩L2(Rd),let (10) ˆ f(ξ) = Zf(x)e−2πιξ·xdx define the Fourier transform. Let (11) µq(x) = X a∈Aq φq(x−a), be the smoothing of the counting measure on Aq. The radius of the atoms of the measure µqis c0/q, with 0 < c0<1 by the choice of φ. Clearly (12) ˆµq(ξ) = X a∈Aq ˆ φ(ξ/q)e−2πιa·ξ, and the function ˆ φis radial. Non-Isotropic Lattice Distance Measures 231 To study the distances between the elements of Aand the origin, define for t > 0: νq,0(t) = ZωK(x/t)dµq(x), Nq,0(t) = ZΩK(x/t)dµq(x) = Zt 0 dνq,0. (13) Above ωKis the Lebesgue measure on ∂K, ΩKis the characteristic function of K. Note that in the first integral µqis actually a Schwartz function, and ωKa distribution. Without loss of generality one can assume that every lattice cube contains exactly one point of A(this can always be achieved for any finite truncation Aqby sparsing it out and subsequent scaling). In this case define the volume discrepancy (14) Eq,0(t) = Nq,0(t)−tdVol K. Studying the quantity Eq,0for the integer lattice has a long history, see [8] for some references. In the general context of homogeneous sets, the quantity Eq,0defined relative to the origin cannot be expected to be smaller in absolute value than O(qd−1). However, averaging with respect to the choice of the center throughout Aqcan result in a nontrivial estimate, important in the context of the Erd¨os distance problem. This issue is briefly discussed further in the paper following (30). The seemingly redundant 0subscripts come from the fact that in the sequel it turns out to be more convenient to work with the weighted quantities (15) [νq(t), Nq(t), Eq(t)] = t1−d 2[νq,0, Nq,0(t), Eq,0(t)]. The quantity νq,0is the density of the measure µqon K-spheres of radius t, centered at the origin. The primitive Nq,0(t) counts the points in K-balls of radius t. Clearly (16) Z∞ 0 νq,0∼qd. By definition of the quantities µqand νq,0, in order to obtain estimates in terms of q, it is legitimate to sample integrals containing νq,0(as well as Eq,0and other versions of νand Eto appear later) by Darboux sums with the step size 1 c1q, for some constant c1. Clearly νq,0vanishes for t > q +q−1, while for t < q −q−1, (17) νq,0≈q−1Γ(t, q−1), 232 A. Iosevich, M. Rudnev where Γ(t, δ) is the number of points of Ain a K-annulus α(t, δ) (defined as (t+δ)K\tK) centered at the origin, with radius tand width δ; further on δwill always be approximately 1 q. More precisely, the statement (17) means that there exist uniform constants c2and c3, such that (18) Γ t, 1 c2q≤νq,0 q≤Γt, c3 q. Let us cover Aqby a set of concentric K-annuli αkaround the origin, all of which have fixed width δ∼1 q. Suppose α1has radius q−1and for k > 1 the inner boundary of αk+1 coincides with the outer boundary of αk. Terminate the construction as soon as qK is covered by the union of αk. Thus k.q2and the K-radii tkof αkgo up to q+O(q−1). Define the annulus standard deviation Dαand the body mean square discrepancy DKas follows: Dα=s1 q2X k Γ2(tk, δ)≈s1 q3Zq 0 ν2 q,0(t)dt, DK=s1 q2X k E2 q,0(tk)≈s1 qZq 0 E2 q,0(t)dt. (19) Theorem 2. Suppose A=Zd. Then (20) Dα, DK.qd−2, d ≥4, Dα, DK.qlog q, d = 3. As far as the weighted quantity νq(t) is concerned, see (15), the estimate (20) of Theorem 2 is tantamount to (21) kνqk2 2=Z∞ 0 ν2 q(t)dt .(qd, d ≥4, q3log2q, d = 3. Theorem 2 implies Theorem 1. Proof of Theorem 1: Assume Theorem 2. By the Cauchy-Schwartz inequality, (22) q2d≈Zq 1 νq,0dt2 ≤ |supp νq,0|Zq 1 ν2 q,0(t)dt, where |supp νq,0|is the Lebesgue measure of the support of νq,0. Substituting the estimates (20) in the right hand side, one gets the lower bound |supp νq,0|&qfor d≥4 and |supp νq,0|&q log2qfor d= 3. Hence, by definition of νq,0, cf. (17), there exists Ω(q2) in d≥4 and Ω(q2/log2q) Non-Isotropic Lattice Distance Measures 233 disjoint K-annuli, of width δ∼1 q, and whose radii do not exceed q, such that each of these annuli contains at least one lattice point. This is equivalent to the statement of Theorem 1. Let us now return to the general homogeneous set Aset-up. Observe that in the same way as (22), as |supp νq,0|.q, for the second moment of the weighted quantity νqone should always have (23) kνqk2 2&qd. Let us write up some integral representations for the quantities νq,Nq. Applying the Plancherel theorem to the integrals in (13), for the weighted quantities (15) we get: νq(t) = td−1 2Zˆ φ(ξ/q)ˆωK(tξ)X a∈Aq e−2πι a·ξdξ, Nq(t) = td+1 2Zˆ φ(ξ/q)ˆ ΩK(tξ)X a∈Aq e−2πι a·ξdξ. (24) Observe that νq(t) extends as zero to t= 0, as well as the fact that the quantity Nq(t) is not in L2(R+) if d= 2. Euclidean case. First let us get an integral representation for the second moment kνqk2 2 when Kis the Euclidean ball, with the notations ωB, ΩBfor the surface and volume measure. In this case the Fourier transform ˆωB(ξ) is radial, namely ˆωB(ξ)∼ kξk1−d 2Jd 2−1(2πkξk), where Jvfurther denotes the Bessel function of order v≥0. Let us skip the factor of 2πin what follows. This can always be accomplished by scaling. After writing the integral (24) for νqin the spherical coordinates we have (25) νq(t)∼√tZ∞ 0 rJd 2−1(rt)ψ(r/q)X a∈Aq Jd 2−1(rkak)dr, where henceforth (26) ψ(r) = ˆ φ(ξ)|kξk=r, so |ψ(r/q)|is asymptotically smaller than any inverse power of r/q. 240 A. Iosevich, M. Rudnev 1. For the principal terms’ contribution into k˜ Ek2 2, omitting uniform positive constants we get (47) X a,b∈Zd\{0} ˆ φ(a/q)ˆ φ(b/q)Z∞ 0 tψ2(t/q)Jd 2(|a|K∗t)Jd 2(|b|K∗t) (|a|K∗bK∗)d 2 dt. The integral in (47), given (a, b) can be rewritten as an integral over Rd: (48) Z[(ΩB◦|a|−1 K∗)∗φq] b (ξ)ξ·[(ΩB◦|b|−1 K∗)∗φq] b (ξ)ξ dξ =Z∇x[(ΩB◦|a|−1 K∗)∗φq](x)·∇x[(ΩB◦|b|−1 K∗)∗φq](x)dx, where, cf. (33), ΩB◦ |a|−1 K∗(x) = |a|−d K∗ΩB(|a|−1 K∗x). The integral in the right-hand side of (48) is clearly zero if ||a|K∗− |b|K∗|>1 q, while if |a|K∗=|b|K∗, it is O(|a|d−1 K∗). Hence, (47) is (49) ≈q∞ X k=1 Γ2 ∗(rk, δ) r2 krd−1 k ψ2(rk/q)≈Z∞ 0 ˜ν2 ∗(t) 1 + t2dt, cf. (34). 2. For the principal a-term and second b-term in the asymptotics (31), omitting uniform constants, we get (50) X a,b∈Zd\{0} ˆ φ(a/q)ˆ φ(b/q)Z∞ 0 ψ2(t/q)Jd 2(|a|K∗t)Jd 2+1(|b|K∗t) |a| d 2 K∗b d 2+1 K∗ dt. Observe that the expression (50) is reminiscent of (35), only in dimension d+ 1. Let wB,WBbe the Lebesgue measure on Sdand the characteristic function of the Euclidean unit ball in Rd+1, respectively. Let (y, ζ)∈Rd+1 ×Rd+1, let the radial cutoff function ϕbe defined in the same way as φ, only in dimension d+ 1. Denote ϕq(y) = qd+1ϕ(qy), wB◦|a|−1 K∗(y) = |a|−d K∗wB(|a|−1 K∗y), as well as WB◦|a|−1 K∗(y) = |a|−d−1 K∗WB(|a|−1 K∗y). Then given (a, b), the integral in (50) is a constant times (51) Z[(wB◦|a|−1 K∗)∗ϕq] b (ζ)·[(WB◦|a|−1 K∗)∗ϕq] b (ζ)kζkdζ ≈Z[(wB◦|a|−1 K∗)∗ϕq](y)·k∇y[(WB◦|a|−1 K∗)∗ϕq](y)kdy. Non-Isotropic Lattice Distance Measures 241 Thus the integral vanishes if ||a|K∗−|b|K∗|>1 qand is approximately 1 |b|d+1 K∗ if |a|K∗=|b|K∗. Summation in absolute values over (a, b) values results precisely in (49). 3. We deal with the remainder in the asymptotics (31) in the same way as it was done in Proposition 2.2. On this step, in the double sum in a, b ∈Zd\ {0}representing the second moment of ˜ Ewe use the leading term for aand the remainder for b. The demonstration consists in essentially repeating (37)–(40). The presence of the cutoff terms ˆ φ(a/q), ˆ φ(b/q) allows here for restricting the summation to Zd q\{0}. Assume |a|K∗≥ |b|K∗and partition each integral into three counterparts in the summation according to (37). For instance, for the first counterpart, cf. (38), we have X a∈Zd q\{0}X b6=0,|b|K∗≤|a|K∗Z|a|−1 K∗ 0 rd+1 dr ≈X a∈Zd q\{0}|a|−2 K∗ ≈(qd−2, d ≥3, log q, d = 2. (52) The estimates of the second and third counterpart are done along the same lines as (39) and (40) and we omit them. Remark. It is clear that estimating the right-hand-side of (46) which is essentially an L2-estimate for ν tdoes not suffice to get a sharp estimate for ν, when it grows on average slower than √tas t→ ∞. That is why we cannot prove Theorem 2 for d= 2. The logarithmic factor in the case d= 3 in the estimate (21) also appears to be an artifact. 4. Proof of Theorem 2 Theorem 2 will follow immediately from the bound (46) of Lemma 3.1 and the following lemma, which somewhat generalizes the results of [8]. Lemma 4.1. We have the following bound: (53) k˜ Ek2 2.bd(q),where bd(q) =        qd−2, d ≥4, qlog2q, d = 3, q, d = 2. 242 A. Iosevich, M. Rudnev Proof: There is no harm changing in (46) the lower limit of integration to 1 and 1 + t2in the denominator to t2. By definition of ˜ν, for any tl,tu, with tu−tl1 qand a small enough δ∼1 q, we have the representation of the integral as a Darboux sum: (54) Ztu tl ˜ν2 ∗(t) t2dt ≈X k ν2 ∗(t)|t∈Ik t2 k ψ(tk/q)δ, where the intervals Ik= [tk, tk+1) of length δpartition [tl, tu) and the choice of t∈[tk, tk+1) is arbitrary. Then one can always choose tinside each interval Ikin such a way that (55) ν∗(t).max htd−1 2, q|E∗|(t)i. Indeed, the first term inside the above maximum corresponds to the case of the existence of t∈Iksuch that ν∗(t).td−1 2. Otherwise, let us use the fact that dE∗(t) dt ≈ν∗(t)+Otd−1 2, and if ν∗(t)&td−1 2, the Otd−1 2term can be omitted. Then |E∗(t)|&Rt t0ν∗(τ)dτ, where at t0,|E∗|has its absolute minimum in Ik. Which implies that qsupIk|E∗(t)|&infIkν∗(t) in this case. Note that due to (23) all the “regular” terms Otd−1 2that appear further would a-priori result in (53), and in fact stronger inequalities for d= 2,3. Furthermore by (55) (56) Z∞ 1 ˜ν2 ∗(t) t2dt .Z∞ 1 td−3ψ(t/q)dt +ZI ˜ν2 ∗(t) t2dt, where (57) I={t:ν∗(t)≤c5q|E∗(t)|}, for some c5. The first integral in (56) bounded via qd−2for d≥3 and log qfor d= 2. Non-Isotropic Lattice Distance Measures 243 Let us turn to the second integral in (56). Clearly, in order to get the upper bound, the integral can be extended from Ito R+, under the assumption that ν∗(t)≤c5q|E∗(t)|everywhere (note that Ican be represented as the union of intervals of length not smaller than ≈1 qeach). Under this assumption, we write out a dyadic decomposition: Z∞ 1 ˜ν2 ∗(t) 1 + t2dt ≈∞ X k=0 2−2kZ2k+1 2k ˜ν2 ∗(t)dt .q∞ X k=0 |ψ(2k/q)|2d+1 4k−2ksZ2k+1 2k|˜ E∗|2ν∗(t)dt. (58) To get the right-hand side we have applied Cauchy-Schwartz and used the fact that in an annulus of width 2kthe integral of ν∗is O(2kd+1 2), recall the scaling (15). Furthermore, using the fact that dE(t) dt ≈ν∗(t) + O(td−1 2), we have (59) Z2k+1 2k|˜ E∗|2ν∗(t)dt .|ψ(2k/q)| |E∗(2k)|3+|E∗(2k+1)|3+2kd−1 2Z2k+1 2k E2 ∗(t)dt!. The cubic terms in brackets are bounded as (60) Oh23k(d−3+2 d+1 )+ 23kd−1 2q−3i, which follows from the well known, see e.g. [10], L∞estimate (61) |E0(t)|.td−2+ 2 d+1 +q−1td−1, where E0(t) = td−1 2E(t), in view of the scaling (15). It is a routine calculation to show using the decay of ψthat the contribution of these terms into (58) is well in compliance with (53). Hence we are left with (62) Z∞ 0 ˜ E2(t)dt .bd(q) + q∞ X k=0 2k(d 2−2)|ψ(2k/q)|sZ2k+1 2k ˜ E2 ∗(t)dt. Assuming that the sum above is &bd(q), see (53), consider the case d≥4 first. Then, as clearly (63) ∞ X k=0 2k(d 2−2)|ψ(2k/q)|.qd 2−2,for d≥4, 244 A. Iosevich, M. Rudnev we have (64) Z∞ 0 ˜ E2(t)dt .qd 2−1sZ∞ 0 ˜ E2 ∗(t)dt, and it follows that k˜ Ek2 2,k˜ E∗k2 2.bd(q) = qd−2,d≥4, as one can certainly swap the subscript ∗to the left-hand side. The case d= 2,3 requires some extra consideration, see [8], which we have adopted from the latter reference for the sake of completeness. Recall that the quantity Ehas been defined with respect to the parameter q, where 1 qis the characteristic scale of the smoothing. To reflect this fact, let us further write E=E(q),˜ E=˜ E(q).It is easy to verify by definition of Ethat for t.¯q.q, one has (65) |E(¯q)|(t).|E(q)|(t) + O(td−1 2¯q−1). Let us rewrite (62) as follows: (66) Z∞ 0|˜ E(q)|2(t)dt .bd(q) +qsup k sR2k+1 0|˜ E(q) ∗|2(t)dt bd(2k+1)  ∞ X k=0 2k(d 2−2)|ψ(2k/q)|qbd(2k+1). Evaluating the sum yields (67) R∞ 0|˜ E(q)|2dt bd(q).1 + sup k sR2k+1 0|˜ E(q) ∗|2dt bd(2k+1) . The supremum above should be achieved for some finite k, because of the decay, built into the quantity ˜ E, due to the presence of the cutoff ψ. Then define ¯ kas follows: (68) m(q) = max sup k R2k+1 0|˜ E(q)|2dt bd(2k+1),R2k+1 0|˜ E(q) ∗|2dt bd(2k+1)  is achieved for k=¯ k. Without loss of generality suppose the maximum in (68) is effected by the first entry. Also suppose m(q)>1, otherwise the proof of Lemma 4.1 would be complete. Non-Isotropic Lattice Distance Measures 245 Then by definition of ¯ k, (69) R∞ 0|˜ E(q)|2dt bd(q).R∞ 0|˜ E(q)|2dt bd(2¯ k+1), hence 2¯ k+1 .q. Consider now two cases, with the objective to show that m(q) = O(1). Case 1: If 2¯ k+1 &q, then the quantity m(q), by its definition, has to be bounded by a constant times the left-hand side of (67). This implies m(q) = O(1). Case 2: Suppose now 2¯ k+1 q, let ¯q= 2¯ k+1. Look back at the expressions (67) and (68) replacing qby ¯q, i.e. as the statements about the quantity ˜ E(¯q)rather than ˜ E(q). By (65), if k=¯ k, then (70) Z2k+1 0|˜ E(¯q)|2dt ≈Z2k+1 0|˜ E(q)|2dt. Otherwise, if k < ¯ k, the relation (70) should in general hold with the .sign. This implies m(¯q)≈m(q), in other words m(¯q) may be thought to be achieved when k=¯ k, so by (70) (71) m(¯q).R∞ 0|˜ E(¯q)|2dt bd(¯q). This, similarly to Case 1, the statement (67) for the quantity ˜ E(¯q), would imply m(¯q) = O(1), so once again m(q) = O(1). Therefore the right-hand side of (67) always turns our to be O(1). This completes the proof of Lemma 4.1 and Theorem 2. Acknowledgements. Research has been partially supported by the NSF Grant DMS02-45369, EPSRC Grant GR/S13682/01 and the Bristol Institute for Advanced Study. References [1] G. Arutyunyants and A. Iosevich, Falconer conjecture, spherical averages and discrete analogs, in: “Towards a theory of geometric graphs”, Contemp. Math. 342, Amer. Math. Soc., Providence, RI, 2004, pp. 15–24. [2] J. Bourgain, Hausdorff dimension and distance sets, Israel J. 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II: planar domains, Preprint (2004). [10] E. Landau,“Vorlesungen ber Zahlentheorie”, Vol. II, S. Hirzel, Leipzig, 1927. [11] P. Mattila, Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets, Mathematika 34(2) (1987), 207–228. [12] W. M¨ uller, On the average order of the lattice rest of a convex body, Acta Arith. 80(1) (1997), 89–100. [13] J. Pach and M. Sharir, Geometric incidences, in: “Towards a theory of geometric graphs”, Contemp. Math. 342, Amer. Math. Soc., Providence, RI, 2004, pp. 185–223. [14] C. D. Sogge,“Fourier integrals in classical analysis”, Cambridge Tracts in Mathematics 105, Cambridge University Press, Cambridge, 1993. [15] G. N. Watson,“A Treatise on the Theory of Bessel Functions”, 2n ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. [16] T. H. Wolff, Decay of circular means of Fourier transforms of measures, Internat. Math. Res. Notices 10 (1999), 547–567. [17] T. H. Wolff,“Lectures on harmonic analysis”, With a foreword by Charles Fefferman and preface by Izabella Laba, edited by Izabella Laba and Carol Shubin, University Lecture Series 29, American Mathematical Society, Providence, RI, 2003. Non-Isotropic Lattice Distance Measures 247 Alexander Iosevich: Department of Mathematics University of Missouri Columbia, MO 65211 USA E-mail address:[email protected]du Misha Rudnev: School of Mathematics University of Bristol Bristol BS8 1TW UK E-mail address:[email protected] Primera versi´o rebuda el 15 de setembre de 2004, darrera versi´o rebuda el 23 de desembre de 2004.