Energy bounds for modular roots and their applications

We generalise and improve some recent bounds for additive energies of modular roots. Our arguments use a variety of techniques, including those from additive combinatorics, algebraic number theory and the geometry of numbers. We give applications of these results to new bounds on correlations between {\it Sali{\'e}} sums and to a new equidistribution estimate for the set of modular roots of primes.

For a prime q we use F q to denote the finite field of q elements.Given a set N ⊆ F q and an integer k 1, let T ν,k (N ; q) be the number of solutions to the equation (in . ., 2ν.For ν = 2 we also denote T ν,k (N ; q) = E k (N ; q).When k = 1, this is the well-known in additive combinatorics quantity called the additive energy of N .More generally, E k (N ; q) is the additive energy of the set of k-th roots of elements of N (of those which are k-th power residues).
The quantity E 2 (N; j, q). has been introduced and estimated in [10].In particular, for any j ∈ F * q , by [10, Lemmas 6.4 and 6.6] we have (1.1)E 2 (N; j, q) min N 4 /q + N 5/2 , N 7/2 /q 1/2 + N 7/3 q o (1) , which has been used in [10,Theorem 1.7] to estimate certain bilinear sums and thus improve some results of [11] on correlations between Salié sums, which is important for applications to moments of L-functions attached to some modular forms.Furthermore, bounds of such bilinear sums have applications to the distribution of modular square roots of primes, see [10,21] for details.
This line of research has been continued in [20] where it is shown that for almost all primes q, for all N < q and j ∈ F * q one has an essentially optimal bound (1.2) E 2 (N; j, q) N 4 /q + N 2 q o (1) .
As an application of the bound (1.2), it has been show in [20] that on average over q one can significantly improve the error term in the asymptotic formula for twisted second moments of L-functions of half integral weight modular forms.Furthermore, it is shown in [20] that methods of additive combinatorics can be used to estimate E 2 (N ; q) for sets N with small doubling.Namely, for an arbitrary set N (of any algebraic domain equipped with addition), as usual, we denote Then it is shown in [20], in particular, that if N ⊆ Z q is a set of cardinality N such that # (N + N ) LN for some real L, then Here we extend and improve these results in several directions and obtain upper bounds on T ν,k (N ; q) and T ν,k (N; j, q) for other choices of (ν, k) besides (ν, k) = (2, 2) along with improving the bound of [10,Lemma 6.6] for T 2,2 (N; j, q).
Our estimate for T 2,2 (N; j, q) gives some improvement on exponential sums bounds from [10].Obtaining nontrivial bounds on T ν,k (N; j, q) with ν > 2 have a potential to to obtain further improvements and extend the region in which there are non-trivial bounds of bilinear sums from [10,20].In turn this can lead to further advances in their applications.
One such application is to bilinear sums with some multidimensional Salié sum which by a result of Duke [9] can be reduced to one dimensional sums over k-th roots (generalising the case of k = 2, see [15][ Lemma 12.4] or [18][ Lemma 4.4]).This result of Duke [9] combined with our present results and also the approach of [10,11,20], may have a potential to lead to new asymptotic formulas for moments of L-functions with Fourier coefficients of automorphic forms over GL(k) with k 3. 1.2.Notation.Throughout the paper, the notation U = O(V ), U ≪ V and V ≫ U are equivalent to |U| cV for some positive constant c, which throughout the paper may depend on the integer k.

For any quantity
For complex weights β = {β n } n∈N , supported on a finite set N , we define the norms , where σ > 1, and similarly for other weights.For a real A > 0, we write a ∼ A to indicate that a is in the dyadic interval A/2 a < A.
We use #A for the cardinality of a finite set A.
Given two functions f, g on some algebraic domain D equipped with addition, we define the convolution We can then recursively define longer convolutions ( If f is the indicator function of a set A then we write In fact, we often use A(a) for the indicator function of a set A, that is, A(a) = 1 if a ∈ A and A(a) = 0 otherwise.
Note that (A • A) (d) counts the number of the solutions to the equation d = a 1 − a 2 , where a 1 , a 2 run over A, that is As usual, we also write Finally, we follow the convention that in summation symbols a A the sum is over positive integers a A.
We see that Theorem 1.2 is sharper than (1.5) provided N q 1/16 .The proofs of Theorem 1.1 and Theorem 1.2 are based on the geometry of numbers and in particular on some properties of lattices.
Next we generalise (1.2) to higher order roots.In fact, as in [20] the methods allow us to also treat the natural extension of E k (N; j, q) to composite moduli q, for which we consider equations in the residue ring Z q modulo q, and estimate E k (N; j, q) for almost all positive integers q.We however restrict ourselves to the case of prime moduli q.
To establish Theorem 1.3 we use some arguments related to norms of algebraic integers.
We now extend the bound (1.3) to other values of k as follows.
Theorem 1.4.Let N ⊆ F q be a set of cardinality #N = N q 2/3 such that # (N + N ) LN for some real L. Then for k 3 we have 1) , where We remark that the exponent of L in Theorem 1.4 is ϑ 3 = 32/19 and 103 for k 5.For k = 4 the exponent of L is better than generic because of some additional saving in our application of the Plünnecke inequality, see [24,Corollary 6.29].
The proof is based on some ideas of Gowers [12,13], in particular on the notion of the Gowers norm.Finally, we remark that it is easy to see that, actually, our method works for any polynomial not only for monomials.Also, it is possible, in principle, to insert the general weight β but the induction procedure requires complex calculations to estimate this more general quantity Nevertheless, we record a simple consequence of Theorem 1.4 with weights β, which follows from the pigeonhole principle.
Corollary 1.5.Let N ⊆ F q be a set of cardinality #N = N such that # (N + N ) LN for some real L. Then for any weights β supported on N , and with β ∞ 1 Then 1) , where ϑ k and ρ k are as in Theorem 1.4.
We also remark that Theorem 1.4 can be reformulated as a statement that for any set A ⊆ F q either the additive energy #{a 1 + a 2 = a 3 + a 4 : a 1 , a 2 , a 3 , a 4 ∈ A} of A is small or A k has large doubling set
Applying Theorem 1.1, we obtain the following bound.
We now give a new bound for V a,q (α; h, M, N).This does not rely on energy estimates although may be of independent interest.It is also used in a combination with Corollary 2.1 to derive Theorem 2.3 below.Theorem 2.2.For any positive integers M, N satisfying MN ≪ q and M < N, any weight α satisfying α ∞ 1, and a function ϕ satisfying (2.2), we have Corollary 2.1 may be used to improve various results from [10, Sections 1. 3-1.4].We present once such improvement to the distribution of modular roots of primes.Recall that the discrepancy D(N) of a sequence in ξ 1 , . . ., ξ N ∈ [0, 1) is defined as For a positive integer P we denote the discrepancy of the sequence (multiset) of points {x/q : x 2 ≡ p mod q for some prime p P } by Γ q (P ).Combining the Erdös-Turán inequality with the Heath-Brown identity reduces estimating Γ q (P ) to sums of the form (2.1) and (2.3).Combining, Corollary 2.1 with Theorem 2.2, we obtain an improvement on [10,Theorem 1.10].
Note that Theorem 2.3 is nontrivial provided P q 13/22 and improves on the range P q 13/20 from [10, Theorem 1.10].
3. Proof of Theorem 1.1 3.1.Lattices.We use Vol(B) to denote the volume of a body B ⊆ R d .For a lattice Γ ⊆ R d we recall that the quotient space R d /Γ (called the fundamental domain) is compact and so Vol(R d /Γ) is correctly defined, see also [24,Sections 3.1 and 3.5] for basic definitions and properties of lattices.In particular, we define the successive minima λ j , j = 1, . . ., d, of B with respect to Γ as λ j = inf{λ > 0 : λB contains j linearly independent elements of Γ}, where λB is the homothetic image of B with the coefficient λ.
The following is Minkowski's second theorem, for a proof see [24,Theorem 3.30 where then elementary algebraic manipulations imply . ., N}.Since for any λ, µ ∈ F q the number of solutions to , we derive from (3.1) where This implies where Let L(d) denote the lattice where which follows from the fact that for any distinct points (n 0 , m 0 ), (n 1 .m 1 ) satisfying the conditions in (3.3) we have This implies where Furthermore, if one out of m or n is fixed then the the other number is defined in no more than two ways.
Write (3.7) as Then we see that there are two integers a 1 , a 2 satisfying Hence for each fixed pair (a 1 , a 2 ) there are at most By the Cauchy-Schwarz inequality and a well-known bound on the divisor function, see [15, Equation (1.81)], we now derive Similarly, using (3.8) we obtain Combining (3.9) and (3.10) and substituting into (3.6),we see that Using the bound on the divisor function again we obtain Finally consider S 2 .If d satisfies λ 2 (d) 1 then by Lemma 3.1 and Lemma 3.2 For each |n| 6N there exists at most one value of m satisfying (3.3) and for any two pairs (n 1 , m 1 ), (n 2 , m 2 ) satisfying (3.3) we have Since for any integer r = 0 the bound on the divisor function implies 1) .
By (3.12) which implies  The following is known as a transference theorem and is due to Mahler [16] which we present in a form given by Cassels [7, Chapter VIII, Theorem VI].We apply Lemma 4.1 to lattices of a specific type whose dual may be easily calculated.For a proof of the following, see [5,Lemma 15].Lemma 4.2.Let a 1 , . . ., a d and q 1 be integers satisfying gcd(a i , q) = 1 and let L denote the lattice Then we have Our next result should be compared with the case ν = 3 of [6, Lemma 17].It is possible to give a more direct variant of [6,Lemma 17] to estimate higher order energies of modular square roots (see the proof of Corollary 4.4 below) although this seems to put tighter restrictions on the size of the parameter N. Lemma 4.3.Let q be prime, a, b, c ≡ 0 mod q and L, M, N integers.Let L denote the lattice and let B be the convex body , and λ 1 , λ 2 denote the first and second successive minima of L with respect to B. Then at least one of the following holds: (i) (ii) λ 1 1 and λ 2 > 1.
Hence we have either Clearly (4.4) is the same as (ii).
Next suppose that we have (4.5).By Lemma 3.2, a similar calculation as before, together with (4.3) gives, Applying Lemma 3.1 and using we derive from (4.6) that Let λ * 1 denote the first successive minima of the dual lattice L * with respect to the dual body B * .By Lemma 4.1 The above estimates combined with (4.6) implies Hence, by the definition of λ * Its remains to recall that by Lemma 4.2 and also it is obvious that By (4.7), this implies there exists some λ ≡ 0 mod q and ℓ, m, n satisfying (iii), which completes the proof.
Corollary 4.4.Let ε > 0 be a fixed real number.For j ∈ F * q and integer N ≪ p, let A, D ⊆ F q denote the sets Let K be sufficiently large and suppose K and ∆ satisfy Then either 2 ).Since 0 ∈ D, for each d ∈ D, by (4.9) and [10,Lemma 6.4] there exists The above and (4.12) imply (4.15)K I(f ).
Rearranging (4.14) we obtain q.This implies that I(f ) is bounded by the number of solutions to and B the convex body for a suitable absolute constant C. By (4.13) and (4.16) Let λ 1 , λ 2 denote the first and second successive minima of L with respect to B. Assuming that K 1 we have λ 1 1.
Suppose that Then there exists some (a 0 , b 0 , c 0 ) ∈ L ∩ B such that for any d 1 , d 2 ∈ D satisfying (4.17) we have for some m ∈ Z.Note from (4.13) for each d 1 , d 2 ∈ D we have m d 1 m d 2 = 0 and hence c 0 = 0.This implies Let J(ℓ, m, n) count the number of solutions to for some absolute constant C. We next show that (4.21) J(ℓ, m, n) = N o (1) .
Note we may assume Recall (4.16) and hence from (4.8), assuming that N is large enough, we derive Similarly by (4.24) and (4.25) we have m 2 ≡ nℓ mod q and again (4.8) ensures that m 2 = nℓ.Therefore (4.28) implies the following equation We see that Hence from (4.13) and (4.27), there exists some constant C such that which completes the proof.
We apply Corollary 4.4 to estimate the right hand side of (4.36).We now fix some ε > 0 and suppose first that one of (4.8) or (4.9) does not hold.In particular, assume (4.37) If (4.37) holds, then using the trivial bounds we derive from (4.36) (1) .  (1.

5.
Proof of Theorem 1.3 5.1.Product polynomials.In the proof of [20,Lemma 5.1], a certain polynomial in four variables with integer coefficients played a key role.More precisely, it has been found in [20] that the polynomial has the following property.Letting U = u 2 , V = v 2 , X = x 2 , and Y = y 2 , one has that F (u 2 , v 2 , x 2 , y 2 ) = 0 for any u, v, x, y for which u + v = x + y (over any commutative ring).We now proceed to discuss this property in a more general context.
Denote U k = {ω ∈ C : ω k = 1} and consider the polynomial defined over the cyclotomic field K k = Q (exp(2πi/k)).Since the Galois group Gal(K k /Q) of K is cyclic and any automorphism σ of K k over Q is a multiplication by some ω ∈ U k , we see that Hence G k has rational coefficients.Since obviously these coefficients are algebraic integers, we see that We also see that Thus, it is also a polynomial in X k 1 and of course also in X k 2 and X k 3 .Hence we can write Remark 5.1.It is clear that this construction can be extended in several directions, in particular to polynomials 5.2.The zero set of F k (X 1 , X 2 , X 3 , X 4 ).We now need the following bound on the number of integer zeros of F k in a box.Denote by T k (N) the number of solution to the equation Lemma 5.2.Fix an integer k 3.For any positive integer N, we have Therefore there exist roots of unity ω 1 , ω 2 , ω 3 ∈ U d such that (5.1) We now distinguish two cases.Case 1.At least one of the roots of unity ω 1 , ω 2 , ω 3 is not real.Complex conjugation then provides a second linear equation, which is different from (5.1).Then using (5.1) and (5.2) to eliminate t 4 one obtains a nontrivial linear equation in t 1 , t 2 and t 3 which obviously has at most O(N 2 ) solutions, after which t 4 is uniquely defined.Thus the total number of solutions in Case 1 is O(N 2 ).Case 2. All three of ω 1 , ω 2 , ω 3 are real, that is, ω 1 , ω 2 , ω 3 ∈ {−1, 1}, and the equation (5.1) reduces to We observe that Case 2 also covers the 2N 2 +O(N) diagonal solutions.
To treat the non-diagonal solutions, one can now apply results of Besicovitch [2], Mordell [17], Siegel [22], or the more recent results of Carr and O'Sullivan [8].For instance, [8,Theorem 1.1] shows that a set of real k-th roots of integers that are pairwise linearly independent over the rationals must also be linearly independent.Applying this to the set t 1 , t 2 , t 3 , t 4 , which by (5.3) is not linearly independent over Q, it follows that two of them, for example, t 1 and t 2 , are linearly dependent over Q.We derive that there are positive integers a 1 , a 2 , b such that where b is not divisible by a k-th power of a prime.That is, a k 1 is the largest k-th power that divides n 1 , and a k 2 is the largest k-th power that divides n 2 .
Then letting t 5 denote the positive k-th root of b, the equation (5.3) becomes Without loss of generality, we can assume that a 1 a 2 .Hence for any fixed 1 a 2 a 1 N 1/k there are at most N/a k 1 possible values for b and thus for t 5 .After a 1 , a 2 and t 5 are fixed, there are obviously at most N pairs (t 3 , t 4 ) satisfying (5.4).Hence the total contribution from such solutions is which concludes the proof.
We remark that the case of k = 2 can also be included in Lemma 5.2 however this case is already fully covered by the results of [20].

Concluding the proof. Clearly the congruence
Since for a prime q ∼ Q, a ∈ F q and j ∈ F * q , there are at most k solutions to the congruence jz k ≡ a mod q in variable z ∈ F q , and thus at most 2k solution in variable z ∈ Changing the order of summation and separating the sum over the variables U, V, X, Y into two parts depending on whether F (U, V, X, Y ) = 0 or not, we derive Recall that F k is a polynomial with constant coefficients of degree k 3 .Hence F k (U, V, X, Y ) ≪ N k 3 , and thus trivially has at most O (log N) prime divisors.Hence, we derive 1) , and applying Lemma 5.2 we conclude the proof.
Remark 5.3.Furthermore it is easy to see that there is a constant Hence in this range of N, using Lemma 5.2, we obtain E k (N; j, q) ≪ N 2 for every q.
6. Proof of Theorem 1.4 6.1.Preliminary discussion.We need some facts about the Gowers norms, introduced in the celebrated work of Gowers [12,13] on the first quantitative bound for the famous Szemerédi Theorem [23] about sets avoiding arithmetic progressions of length four and longer.As an important step in the proof, Gowers [12,13] observes that there are very random sets having an unexpected number of arithmetic progressions of length l 4.An example is, basically, the set (6.1) where c k > 0 is an appropriate constant, depending on k 2 only (see the beginning of [13,Section 4] and also [14]).Then the set A (k) has an enormous number of arithmetic progressions of length k + 2 but the expected number of shorter progressions.In Theorem 1.4 we consider the sets N 1/k , where N is a set with small doubling.Clearly, such sets generalise the construction (6.1).Below we show that these sets are random in the sense, that they all have small additive energy.Actually, we obtain a stronger property that Gowers norms of its characteristic functions are small and thus this has even more parallels to the Gowers construction (6.1).On the other hand, sets N 1/k preserve all essential We now assume that (6.5) #A s N 4/5 L 32/5 .
We also observe that we can always assume that L N 1/32 as otherwise the result is trivial.Further to show that that the second term in (6.4) dominates the first one, we need to check that (6.6) s (#A s ) 5/4 q, which in turn is equivalent to (#A s ) 3 L 32 N 8 q −4 .Since for L N 1/32 and N q 2/3 we have we see that under the assumption (6.5) we have (6.6) and hence the bound (6.4) becomes (6.7) 1) .By the definition of the sets A s , we have (6.8) Furthermore, using the definition of U 3 -norm we write (6.9) First we observe that Thus for each of E(A) choices (a 1 , a 2 , a 3 , a 4 , s) ∈ A 4 , a 1 + a 2 = a 3 + a 4 there are at most T possibilities for s with #A s T and we derive (6.10) We now choose (6.11) T = 27E(A) −4/5 L 32/5 N 16/5   and note that the trivial upper bound E(A) (#A) 3 27N 3 implies that T N 4/5 L 32/5 .Hence for any s with #A s > T the condition (6.5) is satisfied and so the bound (6.7) holds.
Subtracting the expressions with s and t from the expression with s+t, we see that 3N − 3N contains 12stx We also set T = (E(A)N 2 L 12 A −1 U 3 ) 4/5 and note that we have the trivial bound A U 3 NE(A).We also have T N 4/5 L 48/5 .We now verify that T 3 L 48 N 8 q −4 or N 12/5 L 144/5 L 48 N 8 q −4 which is equivalent to N 28 L 96 q 20 .Since we can clearly assume that L N 1/48 as otherwise the result is trivial, the last inequality hold under our assumption N q 2/3 .Hence, similar to the case k = 3 after simple calculations, one verifies that for #A s,t > T , we have L 2 s,t (#A s,t ) 5/4 q which in turn is equivalent to (#A s,t ) 3 T 3 L 48 N 8 q −4 .Hence, by (1.3) we have Using (6.2) and ( 6.3) and the arguments as above, we get L 48 N 8/5 E 4/5 (A) A 1/5 since again we have chosen T to optimise the above bound.
6.3.3.Case k 5. Finally, consider the general case, which we treat with a version of Weyl differencing.Now and let x ∈ A s 1 ,...,s k−2 .Indeed, we start with A s 1 and reduce the main term in x k , (x + s 1 ) k ∈ N deriving that p k−1 (x) ∈ N − N , where deg p k−1 = k − 1.After that consider (A s 1 ) s 2 = A π(s 1 ,s 2 ) and reduce degree of the polynomial by one, and so on.We also note that by the Plünnecke inequality, see [24,Corollary 6.29 the role of L s or L s,t is now played by We now set Using the same arguments as above, after somewhat tedious calculations to verify all necessary conditions such as (6.15) to obtain 1) .
In particular to check (6.15) we note that for the above choice of T we have T N 4/5 L 2 k+2 /5 , and then derive which is true because N q 2/3 and L N 1/2 k+2 (which we can assume as otherwise the bound is trivial).Using the formula (6.2) and (6.3) we obtain and hence by induction and Lemma 6.2 1) .
In other words, 1) , which completes the proof.
We proceed on a case by case basis depending on the size of N 1 .We first note a general estimate for the multilinear sums.Let I, J ⊆ {1, . . ., J} and write Grouping variables in Σ(V) according to I, J , there exists α, β satisfying α ∞ , β ∞ = Q o (1) , e q (hx).

and let λ 1
(d), λ 2 (d) denote the first and second successive minima of L(d) with respect to B.We now partition summation in (3.2) according to the size of λ 1 (d) and λ 2 (d) to get for some choice of integers m a,b , n a,b satisfying |m a,b |, |n a,b | 6N.Fix some a, b as in the sum in (3.6) and consider K(a, b).If n, m satisfy n 2 − n 2 a,b m − m a,b = a b , |m|, |n| 6N, then, since gcd(a, b) = 1, we have (3.7)n 2 − n 2 a,b ≡ 0 mod |a|, and (3.8) m − m a,b ≡ 0 mod |b|.

4 .
Proof of Theorem 1.2 4.1.Lattices.For a lattice Γ and a convex body B we define the dual lattice Γ * and dual body B * by Γ * = {x ∈ R d : x, y ∈ Z for all y ∈ Γ}, and B * = {x ∈ R d : x, y 1 for all y ∈ B}, respectively.

Lemma 4 . 1 .
Let Γ ⊆ R d be a lattice, B ⊆ R d a symmetric convex body and let Γ * and B * denote the dual lattice and dual body.Let λ 1 , . . ., λ d denote the successive minima of Γ with respect to B and λ * 1 , . . ., λ * d the successive minima of Γ * with respect to B * .For each 1 j d we have λ j λ * d−j+1 d!.

(4. 13 )
Let I(f ) count the number of solutions to the congruence(4.14)

8 . 3 8. 1 .
Proof of Theorem 2.Preliminaries.Our argument follows the proof of [10, Theorem 1.10], the only difference being our use of Corollary 2.1 and Theorem 2.2.We refer the reader to [10, Section 7] for more complete details.Let S q (h, P ) denote the sum S q (h, P ) = and• the arithmetic functions m i → γ i (m i ) are bounded and supported in [M i /2, 2M i ];
The zero set of F k (X 1 , X 2 , X 3 , X 4 ) m d 1 is fixed, due to the coprimality condition in (4.13), n 1 , n 2 satisfying (4.19), , n d 1 n d 2 m d 1 m d 2 = 0. We see from (4.29) that n d 1 m d 1 n d 2 m d 2 =r 2 for some r ∈ Z and hence a bound on the divisor function, see [15, Equation (1.81)], implies and (4.29) holds .Summing the above over f ∈ F , using (4.15) and noting that for each ℓ, m, n satisfying (4.26) there exists O(1) values of f satisfying (4. [24,(t 2 s + ts 2 )x + (t + s) 4 − s 4 − t4and we can apply a version of previous arguments.In particular, since by the Plünnecke inequality, see[24, Corollary 6.29],