SQUARE ROOS AND AND DIRECIONS We onstrut a lattie-like point set in the Eulidean plane that eluidates the relationship between the loal statistis of the frational parts of n and diretions in a shifted lattie. he randomly rotated, and then strethed, point set onverges weakly to a limiting random point proess. his follows by the same arguments as in Elkies and MMullen s original analysis for the gap statistis of n mod [Duke Math. J. 23 (2004, 95 39]. here is, however, a urious nuane: the limit proess is not invariant under the standard SL(2, R-ation on R 2. Let L R 2 be an arbitrary Eulidean lattie of ovolume one. One of the findings in [6] is that the limiting gap distribution for the diretions of points in the shifted lattie L q with q / QL is the same as the Elkies-MMullen distribution for the gaps in the frational parts of ( n n= N, N [4]. he reason why the two limit distributions oinide is that they follow from the equidistribution of two different unipotent translates that onverge to the same invariant measure on the spae of affine latties ASL(2, Z\ ASL(2, R, and are integrated against the same test funtion. We will here illuminate this phenomenon by formulating the Elkies-MMullen onvergene in terms of a natural point proess in R 2 whih is different from the one onsidered in the original paper [4]. he idea is to onstrut a lattie-like point set in R 2 suh that (a its diretions exatly reprodue the frational parts of n and (b it is approximated loally (in a suitable sense by affine latties. he relevant estimates are immediate from Elkies and MMullen s work. Put ( D( = ( /2 0 0 /2, k(θ = ( os θ sin θ sin θ os θ and let θ be a random variable on R/2πZ distributed aording to an absolutely ontinuous probability measure λ. For fixed L, q as above, and > 0, the random set (2 Ξ = (L qk(θd( defines a point proess in R 2. (Elements in R 2 are viewed as row vetors. A key observation of [6] is that the onvergene of the gap distribution for the diretions follows from the weak onvergene of the point proesses (3 Ξ Ξ (, where Ξ is distributed aording to the unique ASL(2, R-invariant probability measure on the spae of affine latties. We realize ASL(2, R as the semidiret produt SL(2, R R 2 with multipliation law (M, ξ(m, ξ = (MM, ξm + ξ. he (right ation of (M, ξ on R 2 is defined by x xm + ξ, and the spae of affine latties Date: 29 September 205. he researh leading to these results has reeived funding from the European Researh Counil under the European Union s Seventh Framework Programme (FP/2007-203 / ERC Grant Agreement n. 2947.,
2 SQUARE ROOS AND AND DIRECIONS Figure. he point set P. an be identified with the homogeneous spae X := Γ\G with G = ASL(2, R, Γ = ASL(2, Z. We embed SL(2, R in ASL(2, R by M (M, 0 and will also denote the image by SL(2, R. Let us turn to n mod. Consider the point set (4 P = {( n π os ( 2π n n, π sin ( 2π n n N}, see Figure, and the orresponding random point set (5 Ξ = Pk(θD(, see Figure 2. Note that P has uniform density in R 2. hat is, for any bounded A R 2 with boundary of Lebesgue measure zero, (6 lim #(P A 2 = vol A. his follows from the fat that ( n n N is uniformly distributed modulo one (approximate A by finite unions and intersetions of setors of varying radii.
SQUARE ROOS AND AND DIRECIONS 3 Figure 2. A realization of Ξ with = 4 and θ = 0.7. Let H = R <0 R and H + = R 0 R denote the left and right halfplane, respetively. We denote by Γ 2,0 (4 the ongruene subgroup { ( ( } 0 2 (7 Γ 2,0 (4 = M SL(2, Z : M or mod 4. 0 0 Define the random point set (8 Ξ = ( Z 2 g H + ( ([ Z 2 + ( 2, 4 ] g H, where g is distributed aording to the unique G-invariant probability measure µ on the homogeneous spae Y := Λ\G with Λ = Γ 2,0 (4 Z 2. Note that (9 ( 2, 4 Γ 2,0(4 ( 2, 4 + Z2, and hene [ Z 2 + ( 2, 4 ] g is independent of the hoie of representative of g in Y. he purpose of this note is to prove the following. heorem. If θ is random aording to an absolutely ontinuous probability measure on R/2πZ, then Ξ Ξ as. We will see below (heorem 5 that Ξ is not invariant under the standard SL(2, R ation on R 2, although its two-point funtion is the same as that of a Poisson proess
4 SQUARE ROOS AND AND DIRECIONS Figure 3. A realization of the lattie (40 orresponding to = 4 and ξ = 2π θ with θ = 0.7. (heorem 4. Similar proesses arise in the Boltzmann-Grad limit of the Lorentz gas in polyrystals [8]. he plan for the proof of heorem is to show that, in any bounded set A R 2, the set Ξ is very lose to an affine lattie, and then apply a slight extension of the Elkies-MMullen equidistribution theorem, whih we will state now. Let (0 N(ξ = Equidistribution ( ( 2ξ, ( ξ, ξ 2. 0 Note that R G, ξ N(ξ, defines a group homomorphism. heorem 2 (after Elkies-MMullen [4]. Let λ be an absolutely ontinuous probability measure on. hen, for any bounded ontinuous funtion f : X X R, ( lim f ( ΓN(ξD(, Γ(, ( 2, 4 N(ξD( dλ(ξ = f ( Γg, Γ(, ( 2, 4 g dµ(g. Y
SQUARE ROOS AND AND DIRECIONS 5 Figure 4. Figures 2 and 3 superimposed. his illustrates the approximation of Ξ by an affine lattie in fixed bounded subsets of the right halfplane. Proof. he spae Y is a finite over of X. he equidistribution theorem of [4] extends to Y (and in fat to any quotient by a finite-index subgroup of Γ: For any bounded ontinuous funtion h : Y R, we have (2 lim h ( ΛN(ξD( dλ(ξ = Y h(gdµ(g. o omplete the proof, hoose h(g = f ( Γg, Γ(, ( 2, 4 g and note that due to (9 this h is indeed well-defined as a funtion on Y. Properties of the limit proess For f C(R 2 R 2 with ompat support, define the two-point funtion R ± 2 : Y R by (3 R + f (g = m =m 2 Z 2 f (m g, m 2 g, (4 R f (g = m,m 2 Z 2 f (m g, m 2 (, ( 2, 4 g. his defines a linear funtional from L (R 2 R 2 to L (Y, dµ.
6 SQUARE ROOS AND AND DIRECIONS Figure 5. A realization of the lattie (47 orresponding to = 4 and ζ = θ+π 2π with θ = 0.7. heorem 3. For f L (R 2 R 2, (5 Y R ± f R (gdµ(g = f (x, ydxdy. 2 R2 Proof. he relation for R + 2 is proved in [3, Proposition A.3]. he other is analogous: We may assume w.l.o.g. f C(R 2 R 2 of ompat support (use density in L and Lebesgue monotone onvergene theorem. Let µ 0 denote the SL(2, R-invariant probability measure on Y 0 := Γ 2,0 (4\ SL(2, R. hen (6 R f (gdµ(g Y = f ((m + ym, (m 2 + y + ( Y 0 2 2, 4 Mdydµ 0(M m,m 2 = f (ym, (m + y + ( 2, 4 Mdydµ 0(M = Y 0 R 2 m Y 0 R 2 m f (y, (m + ( 2, 4 M + ydydµ 0(M.
SQUARE ROOS AND AND DIRECIONS 7 Figure 6. Figures 2 and 5 superimposed. his illustrates the approximation of Ξ by an affine lattie in fixed bounded subsets of the left halfplane. he Siegel integral formula for f C(R 2 with ompat support reads (7 f ((m + ( 2, 4 Mdµ 0(M = f (xdx. Y 0 m his follows either by diret omputation or the general Siegel-Veeh formula []; it is also a speial ase of the generalized Siegel-Veeh formula for Eulidean model sets (a speial lass of quasirystals [7, heorem 5.]. Applying this with R 2 (8 f (x = f (y, x + ydy R2 proves the theorem. he two-point funtion of the random point proess Ξ is defined by (9 K f ( Ξ = E f (y, y 2. y =y 2 Ξ he following orollary of heorem 3 shows that the the two-point funtion of Ξ is the same as that of a Poisson proess.
8 SQUARE ROOS AND AND DIRECIONS heorem 4. For f L (R 2 R 2, (20 K f ( Ξ = f (x, x 2 dx dx 2. R 2 R 2 Proof. Let (2 f ++ (x, x 2 = f (x, x 2 χ H+ (x χ H+ (x 2, (22 f + (x, x 2 = f (x, x 2 χ H+ (x χ H (x 2, (23 f + (x 2, x = f (x, x 2 χ H (x χ H+ (x 2, (24 f (x, x 2 = f (x, x 2 χ H (x χ H (x 2, be the restritions to the various halfplanes. We have (25 K f ( Ξ = Y [ R + f ++ (g + R + f (( 2, 4 g + R f + (g + R f + (g ] dµ(g. he orollary now follows from heorem 3 by integrating term-wise. Given a subgroup H G, we say Ξ is H-invariant if Ξh has the same distribution as Ξ (viewed as random point sets in R 2 for all h H. Consider the subgroup {( ( } a b (26 P =, (0, y : b, y R, a R \ {0} G. 0 /a heorem 5. Ξ is P-invariant but not SL(2, R-invariant. Proof. he invariane under (27 ( ( a b, (0, y 0 /a is evident for all a > 0, b R. he invariane under (, (0, 0 follows from the fat that the probability of having at least one point on the vertial axis is zero. his proves the P-invariane. As to the SL(2, R-invariane, onsider the set (28 A δ = ([0.75, 2.25] [ 2.25, 2.25] \ (Z 2 + D δ where D δ is a disk of radius δ entered at the origin. he laim follows from the fat that there exists a δ > 0 suh that ( 0 (29 P(A δ Ξ = > 0 and P(A δ Ξ = = 0. 0
SQUARE ROOS AND AND DIRECIONS 9 Proof of heorem We now turn to the proof of heorem. As mentioned before, the following derivation is almost idential to [4], where A is the triangle {(x, y : 0 < x <, y σx}. Note first of all that {( } n n (30 Ξ = π os(2π n θ, π sin(2π n θ : n N. It is suffiient to prove the theorem for retangles of the form A = [a, b] [, d]. Assume for now a > 0. he ase b < 0 is similar, and is disussed below. More general sets A (with boundary of Lebesgue measure zero an then be approximated by unions of suh retangles. For large, the sine has to be small, and thus the argument must be lose to 0 or π. If it is lose to π, the osine is negative whih is ruled out by the assumption a > 0. hus (3 2 πn n θ ( 2π + m + O (n 3/2 d 2 πn for some m Z (whih is unique for suffiiently large in terms of, d. ξ = θ/2π. he inequality transforms to (32 2 πn (m + ξ ( d n + O (n 3/2 2 (m + ξ, πn whih is equivalent to [ ] 2 ( (33 2 (m + ξ n + O πn n 3/2 Now (33 is equivalent to ( (m + ξ (34 + (m + ξ 2 n + O πn n 3/2 Using n = (m + ξ + O(/ n we get that ( (35 + (m + ξ 2 n + O π [ d 2 πn ] 2 (m + ξ. d(m + ξ + (m + ξ 2. πn d π + (m + ξ 2. Let us assume that ξ / [ δ, δ] + Z, where δ > 0 may be hosen arbitrarily small. his assumtion is w.l.o.g., beause the event ξ [ δ, δ] + Z has probability at most λ([ δ, δ], whih tends to zero as δ 0. We an then drop the ondition n, as the positivity of n is implied in (35 for suffiiently large. Next, replaing n by n + m 2 yields (36 π n 2mξ ξ 2 + O ( d π and also (37 a ( π (m + ξ + O b π. Set
0 SQUARE ROOS AND AND DIRECIONS So (38 ( m + ξ, ( π(n 2mξ ξ 2 A + O π and hene ( ( ( 2ξ (39 ( m, n, ( ξ, ξ 2 D(π, 0 A + O 0 We have thus established that in retangles with a > 0 the proess Ξ looks like the affine lattie ( ( ( (40 Z 2 2ξ, ( ξ, ξ 2 D(π, 0, 0 see Figures 3 and 4. Elkies and MMullen have shown that for ξ random with respet to any absolutely ontinuous probability measure on R/Z, (40 onverges weakly to Ξ as. Let us now disuss the remaining ase b < 0. In this ase the argument of the sine has to be lose to π as the osine is now required to be negative. hen (3 is replaed by (4 d 2 πn n θ + π ( 2π + m + O (n 3/2 for some m Z. Set ζ = (θ + π/2π = ξ /2. hus (42 d 2 πn (m + ζ ( n + O (n 3/2 Repeating the steps in the previous alulation leads to (. 2 πn 2 (m + ζ. πn (43 d ( n 2mζ ζ 2 + O π π and (44 a ( π m + ζ + O b π. So (45 ( m + ζ, ( π( n + 2mζ + ζ 2 A + O π and hene ( ( ( 2ζ (46 (m, n, (ζ, ζ 2 D(π, 0 A + O 0 (.
SQUARE ROOS AND AND DIRECIONS We have thus shown that in retangles with b < 0 the proess Ξ looks like the affine lattie ( ( ( Z 2 2ζ, (ζ, ζ 2 D(π, 0 0 ( ( ( ( = Z 2 2ζ, ( ζ, ζ 2 (47 D(π, 0 0 0 [ ( ( ( ] = Z 2 2ζ, ( ζ, ζ 2 D(π, 0. 0 Compare this with Figures 5 and 6. As in the ase of (40, for ζ random, (47 onverges weakly to Ξ as. Individually, Ξ has the same distribution as Ξ, but here we of ourse have to keep trak of the joint distribution with the right halfplane disussed earlier. o this end, note that (48 ( ( 2ζ Γ 0, ( ζ, ζ 2 ( ( = Γ 0 ( ( ( = Γ, ( 2, 4 2ξ 0 ( (, ( 2, 2ξ 4 0, ( ξ, ξ 2, ( ξ, ξ 2 he joint distribution for the number of points in Ξ A and Ξ A 2 with A and A losed retangles properly ontained in the left and right halfplane, respetively, follows from heorem 2 by the arguments of [6]. Using the regularity of the limit distribution as in [6], one an show that retangles of the form [ ɛ, ɛ] [, d] have a vanishing probability of ontaining a random affine lattie point as ɛ 0. his onludes the proof of heorem. Further reading For other aspets of the statistis of n mod we refer the reader to [, 2, 5, 9]; for the diretions in affine latties, see [3, 0]. he latter paper proves an effetive version of Ratner s theorem in this setting..
2 SQUARE ROOS AND AND DIRECIONS Referenes []. Browning and I. Vinogradov, Effetive Ratner theorem for ASL(2, R and gaps in n modulo, arxiv:3.6387 [2] D. El-Baz, J. Marklof and I. Vinogradov, he two-point orrelation funtion of the frational parts of n is Poisson, Pro. AMS 43 (205, 285 2828 [3] D. El-Baz, J. Marklof and I. Vinogradov, he distribution of diretions in an affine lattie: twopoint orrelations and mixed moments. IMRN 205, no. 5, 37 400. [4] N.D. Elkies and C.. MMullen, Gaps in n mod and ergodi theory. Duke Math. J. 23 (2004, 95 39. [5] K. Frazek, R. Shi and C. Uligrai, Generiity on urves and appliations: pseudo-integrable billiards, Eaton lenses and gap distributions, arxiv:508.03946 [6] J. Marklof and A. Strömbergsson, he distribution of free path lengths in the periodi Lorentz gas and related lattie point problems, Annals of Math. 72 (200, 949 2033. [7] J. Marklof and A. Strömbergsson, Free path lengths in quasirystals, Comm. Math.Phys. 330 (204, 723 755. [8] J. Marklof and A. Strömbergsson, Generalized Linear Boltzmann Equations for Partile ransport in Polyrystals. Appl. Math. Res. Express. AMRX 205, no. 2, 274 295. [9] Ya. G. Sinai, Statistis of gaps in the sequene { n}. In Dynamial systems and group ations, volume 567 of Contemp. Math., pages 85 89. Amer. Math. So., Providene, RI, 202. [0] A. Strömbergsson, An effetive Ratner equidistribution result for SL(2, R R 2. Duke Math. J. 64 (205, no. 5, 843 902. [] W. A. Veeh, Siegel measures, Ann. of Math. 48 (998, 895 944. Jens Marklof, Shool of Mathematis, University of Bristol, Bristol BS8 W, U.K. j.marklof@bristol.a.uk