Distributing Points on the Torus via Modular Inverses

We study various statistics regarding the distribution of the points \[\left\{\left(\frac{d}{q},\frac{\overline{d}}{q}\right) \in \mathbb{T}^2 : d \in (\mathbb{Z}/q\mathbb{Z})^{\times}\right\}\] as $q$ tends to infinity. Due to nontrivial bounds for Kloosterman sums, it is known that these points equidistribute on the torus. We prove refinements of this result, including bounds for the discrepancy, small scale equidistribution, bounds for the covering exponent associated to these points, sparse equidistribution, and mixing.

where d ∈ (Z/qZ) × is the multiplicative inverse of d, so that dd ≡ 1 (mod q). There are ϕ(q) such points; they are the image in T 2 of the modulo q hyperbola d, d ∈ ((Z/qZ) × ) 2 : dd ≡ 1 (mod q) .
Associated to this set of points is the probability measure µ q on T 2 defined by These measures equidistribute on T 2 as q → ∞ [BK02,Zha96] (see also [EL18]), so that denotes the Kloosterman sum, and so the Weil bound for Kloosterman sums (see (3.2)) implies that this integral is O m,n (τ (q) √ q/ϕ(q)) for (m, n) = (0, 0), from which equidistribution follows.
(Note, however, that S(m, n; q)/ϕ(q) can be much larger should m and n vary with q; in particular, if m, n ≡ 0 (mod q), then this is equal to 1.) In this paper, we study various refinements of this equidistribution result; we refer to the survey of Shparlinski [Shp12] for further possible refinements and generalisations in different directions. Our emphasis is on quantifying in various ways how the measures µ q behave like analogous measures associated to random points. This is motivated by recent work of Bourgain, Rudnick, and Sarnak [BRS17], where analogous refinements of equidistribution are studied in the setting of lattice points on the sphere, namely statistics in the large n limit of the projection onto the unit sphere S 2 ⊂ R 3 of the set of (x 1 , x 2 , x 3 ) ∈ Z 3 : x 2 1 + x 2 2 + x 2 3 = n .
1.2. Discrepancy. Our first refinement is bounding the discrepancy of the measures µ q as q → ∞. The ball discrepancy is the quantity Here the supremum is over all injective geodesic balls B R (y) in T 2 . This is distinct from the box discrepancy D box (µ q ), where instead of taking a supremum over balls B R (y), one instead takes a supremum over all boxes It is natural to conjecture that D(µ q ) ε q −1/2+ε , since this is the case for random points. We make partial progress towards this conjecture, while also showing that for any fixed δ > 0, the bound |µ q (B R (y)) − vol(B R )| q −1/2+δ is valid for almost all centres y ∈ T 2 of balls B R (y).
This should be compared to the related work of Lubotzky, Phillips, and Sarnak on Hecke orbits of points on the sphere, where square-root cancellation of the spherical cap discrepancy is conjectured and the bound O(q −1/3 (log q) 2/3 ) is proven [LPS86, Conjecture 2.4 and Theorem 2.5]. On the other hand, there exist number-theoretic situations where square-root cancellation is not possible; Jung and Sardari have recently shown that the discrepancy associated to Hecke operators for modular forms of weight k is Ω(k −1/3 (log k) −2 ) [JS20, Theorem 1.1].
1.3. Small Scale Equidistribution. We next consider small-scale equidistribution, namely the shrinking target problem in which one aims to show that µ q (B q )/ vol(B q ) → 1 for a sequence of sets B q whose volume shrinks as q grows. By a pigeonhole-principle argument, we cannot always expect equidistribution in shrinking balls B R (y) ⊂ T 2 of radius R for which R = o(ϕ(q) −1/2 ). Moreover, there are specific regions where we never find any points: if d ∈ (Z/qZ) × with 2 ≤ d ≤ √ q, then d > √ q, so that µ q (B R (y)) = 0 for R = 1/2 √ q and y = (1/2 √ q, 1/2 √ q).
Nonetheless, it is natural to expect that equidistribution holds down to the optimal scale, so that for any fixed δ < 1/2, we have that µ q (B R (y)) ∼ vol(B R ) for fixed y ∈ T 2 and for all q −δ < R < 1/2. Such an asymptotic formula holds for random points. The case y = (0, 0) would then imply a folklore conjecture on the existence of small modular inverses (see [Gar06]), while the case y = (1, 0) would imply a slightly weaker form of a conjecture of Ford, Khan, Shparlinski, and Yankov on the maximal difference of modular inverses [FKSY05, Conjecture 4.2].
Much like for the discrepancy, we are able to make partial progress towards these conjectures, as well as prove an optimal result for almost all centres of balls.
1.4. Covering Exponents. The covering radius R(P n ) of a set of points P n = {x 1 , . . . , x n } ⊂ T 2 is the least R > 0 for which every point y ∈ T 2 is within distance at most R of some point x j in P n . A packing argument implies that the covering radius of any set of n points cannot be o(1/ √ n). The covering exponent of a sequence P of sets of points P n ⊂ T 2 is the quantity .
Closely related to this is the average covering exponent of a sequence P = {P n } of sets of points in T 2 . We letR(P n , δ) be the least R > 0 for which the measure of the set of points y ∈ T 2 not within distance R of a point in P n is at most R −δ . The average covering exponent of P is .
(1) The covering exponent of the sequence of sets of points S q ⊂ T 2 in (1.1) is at most 2.
(2) The average covering exponent of the sequence of sets of points S q ⊂ T 2 in (1.1) is 1.
This should be compared to [BRS17, Section 1.4] and [HR19, Section 1.1] for analogous results for lattice points on the sphere S 2 , and to [Sar19a, Theorem 1.8 and Corollary 1.9] and [Sar19b, Corollary 1.6] for lattice points on higher-dimensional spheres S d with d ≥ 3.
1.5. Variance Bounds and Asymptotics. Theorems 1.5 (2) and 1.6 (2) are consequences of essentially sharp upper bounds for the variance It is natural to conjecture that for all R q −δ for some fixed δ > 0, we have that since such an asymptotic holds for random points. The analogous statement for lattice points on the sphere is [BRS17, Conjecture 1.7]. A modification of this conjecture, replacing balls with annuli, was partially resolved by the author and Radziwi l l [HR19, Theorem 1.3]. We are able to resolve this conjecture provided that q is prime and R is sufficiently small, namely such that R √ q log q → 0; note that generically µ q (B R (x)) = 0 in this regime, so this should be thought of as the "trivial" regime.
(1) For any R < 1 2 and for all ε > 0, the variance (1.8) satisfies (2) Let q be prime. For any R < 1 2 , the variance (1.8) satisfies 1.6. Sparse Equidistribution. We next consider the problem of sparse equidistribution, where we replace the measures µ q with those associated to small subsets of (Z/qZ) × . Of course, equidistribution fails for subsets such as {a ∈ (Z/qZ) × : a ≤ q/2}, since the corresponding measure is supported on {(x, y) ∈ T 2 : x ≤ 1/2}. For this reason, we restrict our study to subsets with algebraic structure, namely cosets in (Z/qZ) × . For each subgroup H q of (Z/qZ) × and corresponding coset aH q ⊂ (Z/qZ) × (so that a = 1 corresponds to the subgroup itself), we define the probability measure µ aHq on T 2 by We prove the following.
Theorem 1.10. Fix δ > 0. For each positive cubefree integer q, pick a subgroup H q of (Z/qZ) × and an associated coset aH q ⊂ (Z/qZ) × for which #H q q 1 2 +δ . Then the probability measures µ aHq equidistribute on T 2 as q tends to infinity along cubefree integers. Furthermore, the same holds only under the assumption #H q q δ provided that q tends to infinity along primes.
It is natural to conjecture that equidistribution holds under the weaker assumption #H q q δ for all positive integers q, not just primes (though cf. Remark 3.10). This is the analogue of [MiVe06, Conjecture 1], in which Michel and Venkatesh pose a similar conjecture for the equidistribution of subsets of Heegner points indexed by small subgroups of the class group of an imaginary quadratic field (see also [HM06] and, more generally, [Ven10]). Michel and Venkatesh note that in their setting, the generalised Lindelöf hypothesis implies a result analogous to Theorem 1.10. 1.7. Mixing. Finally, we consider the problem of mixing. The group (Z/qZ) × acts on the set For each a ∈ (Z/qZ) × , we let We associate to this a probability measure µ q;a on T 4 = T 2 × T 2 via We are interested in the limiting behaviour of these probability measures.
Theorem 1.11. The probability measures µ q;a equidistribute on T 4 as q tends to infinity along primes if and only if a and q − a both tend to infinity with q.
This is the analogue of the mixing conjecture of Michel and Venkatesh on the joint equidistribution of Heegner points [MiVe06], which has been conditionally resolved by Khayutin [Kha19].

Tools
To begin, we let k : R 2 × R 2 → R be a point-pair invariant, so that k(x + w, y + w) = k(x, y) for all x, y, w ∈ R 2 , which gives rise to a point-pair invariant K : T 2 × T 2 → R given by For R > 0, we take k = k R given by and let K = K R denote the associated point-pair invariant on T 2 × T 2 ; if R < 1/2, then this is the indicator function of B R (y). We may calculate the Fourier coefficients of K R as follows: Then for 0 < ρ < R, we define It is readily checked that k ± R,ρ (x, y) are both nonnegative, continuous, pointwise linear in radial coordinates, bounded by 1, and satisfy Thus for all x, y ∈ T 2 , we have the pointwise inequalities Moreover, the Fourier coefficients are given by (2.5) otherwise; in particular, the Fourier series for K ± R,ρ converges absolutely. This follows from the bound
Proof of Theorem 1.9 (2). Upon opening up the Kloosterman sums in (3.5), the variance is equal to the absolutely convergent spectral expansion The diagonal terms, namely those for which d 1 = d 2 , contribute by Parseval's identity, recalling (2.3). The off-diagonal terms, namely those for which d 1 = d 2 , break up into the sum of four separate terms dependent on the following conditions on m and n: (1) m ≥ 1 and n ≥ 0, (2) m ≤ 0 and n ≥ 1, (3) m ≤ −1 and n ≤ 0, (4) m ≥ 0 and n ≤ −1. We observe that the third term is equal to the first term and the fourth term is equal to the second term by reindexing (m, n) with (−m, −n) and (d 1 , d 2 ) with (−d 1 , −d 2 ). Upon additionally reindexing (m, n) with (−n, m) and (d 1 , d 2 ) with (d 1 , d 2 ) for the second term, we deduce that the contribution from the off-diagonal terms is equal to the absolutely convergent expression where we have used the fact that e(x) + e(−x) = 2 cos(2πx). We now use partial summation on both the sum over 1 ≤ m ≤ X and the sum over 0 ≤ n ≤ Y in (3.7), so that the inner double sum over m and n in (3.7) is equal to We shall show that the last three terms in (3.8) converge to 0 as X, Y → ∞ and the first term is absolutely convergent as X, Y → ∞; we shall then bound the limit of this first term. By evaluating these geometric series, we have the bounds independently of x and y.
From this, the last three terms in (3.8) tend to zero as X, Y → ∞. For the first term in (3.8), we observe that x 2 + y 2 dx dy R 2 by first making the change of variables x → x/2πR and y → y/2πR and then once more using [GR15,8.440 x 2 + y 2 2 x 2 + y 2    xy for x 2 + y 2 1, xy (x 2 + y 2 ) 5/2 for x 2 + y 2 1.
It follows that the contribution from the off-diagonal terms (3.7) is bounded in absolute value by a constant multiple of and we have used the fact that q is prime to ensure that c 1 , c 2 ∈ (Z/qZ) × . We have that S(c 1 , c 2 ) = # d 1 ∈ (Z/qZ) × : d 2 1 − c 1 d 1 + c 1 c 2 ≡ 0 (mod q) , and so S(c 1 , c 2 ) ≤ 2 since the quadratic congruence d 2 1 − c 1 d 1 + c 1 c 2 ≡ 0 (mod q) has at most two solutions modulo a prime q (see, for example, [KL13, Lemma 9.6]). We conclude that the contribution from the off-diagonal terms is bounded in absolute value by a constant multiple of where the last inequality follows via the same method as the proof of the Pólya-Vinogradov inequality; see [MoVa07,.
Proofs of Theorems 1.7 (1) and (2). Theorem 1.7 (1) follows immediately from Theorem 1.6 (1). Similarly, the upper bound for the average covering exponent in Theorem 1.7 (2) follows immediately from Theorem 1.6 (2), while the lower bound is a simple consequence of a packing argument.
Proof of Theorem 1.10. For each pair of integers m, n ∈ Z, we have that md + nd q via character orthogonality. The sum over d ∈ (Z/qZ) × is, by definition, the twisted Kloosterman sum S χ (m, n; q). Since the number of characters χ modulo q for which χ| Hq = 1 is ϕ(q)/#H q , the proof then follows from the Weil bound S χ (m, n; q) ≤ τ (q) (m, n, q)q, which is known to hold for cubefree q via [KL13, Propositions 9.4, 9.7, 9.8, and Lemma 9.6]. When q is prime, we instead note that since H q must be cyclic, we may write with f (x) = a 1 x k 1 + a 2 x k 2 for a 1 = am, a 2 = an, k 1 = (q − 1)/#H q , and k 2 = (q − 1)(#H q − 1)/#H q . Since q is a large prime and m, n are fixed, we may assume without loss of generality that m and n are coprime to q. As and #H q q δ by assumption, the conditions of [Bou05, Theorem 1] are met, which allows us to conclude that there exists δ > 0 such that 1 #H q d∈aHq e md + nd q q −δ .
Remark 3.10. When q is not cubefree, it is known that the Weil bound for S χ (m, n; q) may fail; see [KL13, Example 9.9]. For this reason, it is conceivable that the extension of Theorem 1.10 to arbitrary q is false when q is not cubefree.
Proof of Theorem 1.11. It suffices to show that each for fixed (m, n, m , n ) ∈ Z 4 , S m + am , n + an ; q = o m,n,m ,n (q).
The Weil bound for Kloosterman sums (3.2) implies that the left-hand side is bounded in absolute value by 2 (m + am , n + an , q)q. Since q is a large prime and m and n are fixed, we may assume without loss of generality that m and n are coprime to q, in which case the Weil bound gives the desired result unless a ≡ −mm (mod q) and a ≡ −nn (mod q). However, since m, m , n, n are fixed whereas a and q − a tend to infinity with q, these congruences cannot hold for sufficiently large q.
On the other hand, if a does not tend to infinity with q, then there exists a subsequence for which it is equal to a fixed positive integer b by the Bolzano-Weierstrass theorem, and so is equal to 1 along this subsequence, which implies the failure of equidistribution, since An analogous argument shows that equidistribution also fails if q − a does not tend to infinity with q.

Generalisations
Many of the results in this paper can be generalised to the setting studied by Granville, Shparlinski, and Zaharescu in [GSZ05]. Following [GSZ05, Section 2], we take an absolutely irreducible curve C defined over F p embedded in affine space A r (F p ). We may naturally identify C(F p ) with a finite subset of T r via the map F p → T given by x → x/p. Let h = (h 1 , . . . , h s ) : C → A s (F p ) be a suitable rational map that is L-free along C for some L ≥ 0 in the sense of [GSZ05, Section 2]. Then [GSZ05, Theorem 1] states that the probability measures on T s given by f (h(x)) for each measurable function f : T s → C equidistribute on T s as p tends to infinity provided that L tends to infinity with p.
The case of C being the curve y = x and h(x) = (x, x −1 ) corresponds to the equidistribution of the set S q ⊂ T 2 in (1.1) when q = p is prime. The key tool behind [GSZ05, Theorem 1] is a bound for exponential sums due to Bombieri [GSZ05,(8)], which includes the Weil bound for Kloosterman sums of prime level as a special case. Since the key tools for several of the results in this paper are the Weil bound together with bounds for the Fourier coefficients of the indicator function of a ball in T 2 , which of course can be generalised to T s with s ≥ 3, it follows that many of the results in this paper can be generalised to the setting studied by Granville, Shparlinski, and Zaharescu in [GSZ05]. Notably, the discrepancy bound proven in Theorem 1.5 (1) is proven in further generality in [GSZ05, Lemma 3], albeit for the box discrepancy instead of the ball discrepancy.
It is less clear if these methods generalise to curves over Z/qZ with q nonsquarefree, for then the method of Bombieri [GSZ05,(8)] used to bound exponential sums is no longer valid, and instead one must use more elementary methods (see, for example, [IK04, Lemmata 12.2 and 12.3]). Notably, it is not necessarily the case that one can expect bounds in the nonsquarefree setting that are as strong as in the squarefree setting; cf. Remark 3.10.