Bounds for spectral projectors on generic tori

We investigate norms of spectral projectors on thin spherical shells for the Laplacian on generic tori, including generic rectangular tori. We state a conjecture and partially prove it, improving on previous results concerning arbitrary tori.

where χ is a cutoff function taking values in [0, 1] supported in [−1 , 1], and equal to 1 on [− 1 2 , 1  2 ].This definition is understood through the functional calculus for the Laplace-Beltrami operator, which is a self-adjoint operator on (complete) Riemannian manifolds.
A general question is to estimate the exact choice of χ being immaterial 1 .
The answer to this question is known in the case of the Euclidean space: define Then by Stein-Tomas [20,21] we have where we write A ∼ B if the two quantities A and B are such that 1 C A ≤ B ≤ CA, for a constant C which depends only on the d.The answer is again known in the case of compact Riemannian manifolds when δ = 1 (Sogge [19], Theorem 5.1.1),for which 1.2.Spectral projectors on tori.
1.2.1.Different kinds of tori.From now on, we focus on the case of tori given by the quotient , where e 1 , . . ., e d is a basis of R d , with the standard metric.This is equivalent to considering the operators where ∇ is the standard gradient operator, and Q is a quadratic form on R d , with coefficients β ij : Here (β ij ) is a symmetric positive definite real matrix.Dispensing with factors of 2π, which can be absorbed in Q, the associated Fourier multiplier has the symbol χ Q(k) − λ δ .
Standard and rectangular tori correspond to the following particular cases.
• The standard torus corresponds to (e i ) being orthonormal, or β ij = δ ij .
• A rectangular torus corresponds to (e i ) being orthogonal, or equivalently to a diagonal quadratic form β ij = β i δ ij .We will be concerned in this article with generic tori, which for our purposes are defined as follows.
• Consider the rectangular tori with β i ∈ [1,2] for each i; we say a property is true for generic rectangular tori if it is true on a set of (β i ) 1≤i≤d with full Lebesgue measure in [1,2] d .• Consider the tori with β ij = δ ij +h ij for each 1 ≤ i, j ≤ d and some ; we say a property is true for generic tori if it is true for a set of (h ij ) 1≤i≤j≤d with full Lebesgue measure in [− 1 10d 2 , 1 10d 2 ] d(d+1)/2 .1.2.2.The conjecture.It was conjectured in [10] that, for an arbitrary torus, (1.4) P λ,δ L 2 →L p 1 + (λδ) here and below we denote A B if the quantities A and B are such that A ≤ CB for a constant C, where C may depend on the dimension d.This paper also contains new results towards this conjecture, as well as a survey of known results In the present paper, we turn our attention towards generic tori, for which the typical spacing between eigenvalues of CR]; if the β ij are chosen generically we expect these to distribute approximately uniformly.This naturally leads to replacing the above conjecture by the following: for generic tori, (1.5) P λ,δ L 2 →L p β,ǫ 1 + (λδ) where N (λ) is the counting function associated to the quadratic form Q, defined as the number of lattice points To leading order, N (λ) equals Vol(E)λ d , where Vol(E) is the ellipsoid {Q(x) < 1}; the error term is denoted P (λ): For the state of the art regarding P (λ) for any fixed Q we refer the reader to the comments after (1.3) in [10], and to the work of Bourgain-Watt [7] giving an improved bound for the standard two-dimensional torus.For generic quadratic forms, there are a number of additional results.
• It has been shown that the average size of the error, say [E|P (λ 2 ), for different types of averaging: over translations of the integer lattice [15], over shears [14], and over the coefficients (β i ) of a diagonal form [12].
• When d = 2, Trevisan [22] has investigated in more detail the distribution of the normalised error P (λ)λ −1/2 when Q is chosen at random and λ is large.
1.2.4.Known results if p < ∞.After the pioneering work of Zygmund [23], Bourgain [1] asked for L p bounds for eigenfunctions of the Laplacian on the (standard) torus.He conjectured that, if ϕ is an eigenfunction of the Laplacian with eigenvalue λ, then which is equivalent to the case δ = λ −1 of (1.4) for the standard torus.Progress on this conjecture appeared in a series of articles [2,3,4] culminating in the proof of the ℓ 2 -decoupling conjecture by Bourgain and Demeter [5], which implies (1.6) if p ≥ 2(d−1) d−3 and d ≥ 4. Bounds for spectral projectors are essentially equivalent to bounds for the resolvent (−∆ + z) −1 .This was the point of view adopted in Shen [18], Bourgain-Shao-Sogge-Yau [6], and Hickman [11].
Here the goal is to prove a sharp bound when p * = 2d d−2 and δ is sufficiently large.Finally, the authors of the present paper were able to prove the conjecture (1.4) when δ is sufficently large by combining ℓ 2 -decoupling with a geometry of numbers argument [10].
To the best of our knowledge, all works concerned with p < ∞ address either the case of standard tori, or the general case of arbitrary tori; the generic case does not seem to have been considered specifically.This will be a focus of the present paper.

1.3.
A new result through harmonic analysis.The conjecture (1.5) was proved in [10] for arbitrary tori and δ not too small, and for generic tori we can improve the range for δ as follows.
Theorem 1.2.For generic rectangular tori and for generic tori (in the sense of Definition 1.1), the conjecture (1.5) is verified if p > p ST and, for some ǫ > 0, Namely, under these conditions, there holds for almost all choices of (β i ) 1≤i≤d (for generic rectangular tori) or (β ij ) 1≤i,j≤d (for generic tori) In the particularly well-behaved case when p = ∞ and we consider generic diagonal tori, the theorem matches the classical result of Jarník [13] mentioned in the first bullet point in Section 1.2.3, which even promotes the upper bound in the theorem to an asymptotic in that case.
The proof of this theorem will be given in Section 4. The idea of this proof is to first express the spectral projector through the Schrödinger group.First note that the operator χ can also be written χ −Q(∇)−λ 2 λδ by adapting the compactly supported function χ.This in turn can be expressed as and then split off into two pieces, corresponding to |t| λ −1 and |t| λ −1 respectively P λ,δ = λδ χ(λt) χ(λδt)e −2πiλ 2 t e −2πitQ(∇) dt + λδ[1 − χ(λt)] χ(λδt)e −2πiλ 2 t e −2πitQ(∇) dt = P small λ,δ + P large λ,δ .It is easy to see that the operator P small λ,δ can be written in the form δP λ,1 (after adjusting the cutoff function); in other words, this corresponds to the case λ = 1, to which the universal bounds of Sogge apply.
Turning to the term P large λ,δ , its operator norm will be obtained through interpolation between • A bound L p ′ ST → L p ST , Theorem 4.1 below.As noted in [10], is a direct consequence of ℓ 2 decoupling (and valid for any torus).
• A bound L 1 → L ∞ , for which genericity will be used.Namely, we will prove in Section 3 that, generically in (β ij ), (1.7) One could think of (1.7) as square-root cancellation in One could also see this as a minor-arc type bound in the spirit of the circle method; indeed in the case p = ∞ the proof in effect reduces to an application of the Davenport-Heilbronn circle method.
1.4.An elementary approach for p = ∞ and δ very small.When δ is small enough a more elementary counting argument can be used.
Our main result there is Theorem 6.1 below.We first state three particularly simple L 1 → L ∞ bounds proved at the start of Section 6 since they are particularly simple; we will then mention consequences for L 1 → L p bounds.Theorem 1.3.For generic tori, and also for generic rectangular tori, the following holds.If δ < λ 1−2d−ǫ , then We remark that for δ ≤ λ −1 , interpolation with the L p ′ ST → L p ST bound from [10] gives This would always fall short of the conjecture (1.5) for p < ∞, even with an optimal L 1 → L ∞ bound.We highlight a few features of these bounds.
• Although (1.9) and (1.10) do not recover (1.5), they do improve on the best known bounds for N (λ − δ) − N (λ + δ) coming from the results listed in Section 1.2.3.• Both (1.10) and (1.9) are special cases of the stronger estimate (6.1) below, while (1.8) has a short self-contained proof.• We restrict to a > d 2 − 1 in (1.10) and to a ≤ 2d in (1.10) solely becuase the remaining range is already covered by Theorem 1.2 or by (1.8), see also (6.2) below.
• When a = d the bound (1.10) would be trivial, and hence for a ≥ d the bound (1.9) takes over.
In the proof of Theorem 6.1 we will use the Borel-Cantelli Lemma to reduce to estimates for moments of P λ,δ , where the moments are taken over λ and β.A short computation reduces this to the following problem.
Problem 1.4.Estimate (from above, or asymptotically) the number of matrices of the form where the m ij are integers, all entries in each row lie in a specified dyadic range, and also for each k the maximal k × k subdeterminant of P lies in a specified dyadic range.
In section 6.2 we give an upper bound in this counting problem using what is in effect linear algebra, relying on the rather technical Lemma 5.7 below.We are then left with a maximum over all possible choices of the various dyadic ranges, and estimating this maximum will be the most challenging part of the proof.
The bound from Lemma 5.7 could be improved.See Remark 6.2 for one path.Another route concerns the case when all β ij are generic, that is the case of generic tori as opposed to generic rectangular tori.Then one could expand P ; in place of the squares m 2 1i , . . ., m 2 di the ith column would contain all degree 2 monomials in m 1i , . . ., m di .This should allow a smaller bound.

Notations
We adopt the following normalizations for the Fourier series on T d and Fourier transform on R d , respectively: With this normalization, and the Parseval and Plancherel theorems, respectively, are given by The operator m( −Q(∇)) can be expressed as a Fourier multiplier or through a convolution kernel In Sections 5 and 6 we will often join together several matrices A 1 , . . ., A n with the same number of rows to make a larger matrix, for which we use the notation We view column vectors as matrices with one column, so that (A | v) is the matrix A with the vector v added as an extra column on the right.Also in Sections 5 and 6 we use the following notation relating to subdeterminants.If k ≤ min(p, q) and M is a matrix in R p×q , we will denote D k (M ) the maximum absolute value of a k × k subdeterminant of M : We further define D 0 (M ) = 1 for ease of notation, and we let D k (M ) denote the maximal subdeterminant, when the matrix M is restricted to its first ℓ columns: Given two quantities A and B, we denote A B if there exists a constant C such that A ≤ CB, and A ∼ B if A B and B A. If the implicit constant C is allowed to depend on a, b, c, the notation becomes A a,b,c B. In the following, it will often be the case that the implicit constant will depend on β, and on an arbitrarily small power of λ, for instance A β,ǫ λ ǫ B. When this is clear from the context, we simply write A λ ǫ B. Implicit constants in this notation may always depend on the dimension d of the torus that is the object of our study.
Finally, the Lebesgue measure of a set E is denoted mes E.

Bounds on Weyl sums
Consider the smoothly truncated Weyl sum (or regularized fundamental solution for the anisotropic Schrödinger equation) where φ is a smooth cutoff function supported on [−1, 1], equal to 1 on [− 1 2 , 1  2 ].In dimension one, it becomes 3.1.Bound for small time.For small t, the following bound holds on any torus.
where Z decays super-polynomially.
Proof.This is immediate on applying Poisson summation followed by stationary phase.

3.2.
The one-dimensional case.As in, for example, equation (8) of Bourgain-Demeter [3], we have: We now define a decomposition into major and minor arcs: for Q a power of 2, c 0 a constant which will be chosen small enough, and In the definition of Λ Q , the integer a is not allowed to be zero; it turns out to be convenient to single out the case a = 0, by letting Observe that functions of the type Λ Q is the characteristic function of a set: the major acs.
The minor arcs will be the complement, with characteristic function ρ.This gives the decomposition, for any t ∈ R, On the support of each of the summands above, the following bounds are available • On the support of Λ 0 , there holds |t| ≤ 1 N , and we resort to the short time bound.
• On the support of ρ, by Dirichlet's approximation theorem, there exists a ∈ Z and q ∈ {1, . . ., N } relatively prime such that |t − a q | < 1 qN .If q ∼ N , Weyl's bound gives |K 3.3.The case of generic rectangular tori.In this subsection, we assume that the tori are rectangular, or, equivalently, that the quadratic form Q is diagonal.First of all, we learn from the bounds on K N that, on the support of , and ǫ > 0, and for all Q i , N equal to powers of 2, with Proof.Without loss of generality, we can choose β 1 = 1.Indeed, if β 1 , . . ., β d , T γ is changed to γβ 1 , . . ., γβ d , T , with γ > 0, the integral in the statement of the lemma changes by the factor γ. We claim that it suffices to prove that Indeed the case t < 1 4N of the lemma is then immediate, and the remaining case 1 4N ≤ t ≤ N κ follows by the Borel-Cantelli lemma as explained in Appendix A. By definition of Λ Q , To estimate this integral, we observe first that, if Then, by Fubini's theorem Proof.
3.4.The case of generic tori.We start with an averaging lemma.
Proof.If λ ≥ 1, the left-hand side can be bounded by the average of the function min 1 h , N , which equals log N .If λ ≤ 1, the left-hand side is bounded by Armed with this lemma, we can now prove the desired square root cancellation result -its proof already appeared in [8], but we include an equivalent version here for the reader's convenience.Recall that the measure on nonsingular symmetric matrices we consider is given by B = Id +h ij , where h is a symmetric matrix, all of whose coefficients are independent (besides the symmetry assumption) and uniformly distributed in 1 10d 2 , 1 10d 2 .
Lemma 3.6 (Square root cancellation in L 1 t L ∞ x ).Let κ, ǫ > 0; then for generic (β i,j ), there holds, for N a power of 2 and 1 Proof.By the Borel-Cantelli argument in Appendix A, the result would follow from the bound which will now be proved.For x ∈ T d and t ∈ R, applying Weyl differencing gives (where the sum is implicitly restricted to m i , n i having the same parity).By Abel summation, this implies that where Combining the Cauchy-Schwarz inequality with the above yields , where dB = i≤j dβ i,j .We now exchange the order of summation and integration, performing first the integration over B. Without loss of generality, assume that |n 1 | ∼ |n|.Note that tQ 1,1 (n) = t j β 1,j n j ; therefore, by Lemma 3.4, integrating first min t|n| .We integrate next min , giving the same result, and similarly min . Coming back to the sequence of inequalities above,

Proof of Theorem 1.2
An important element of the proof is the optimal L 2 → L p ST bound for spectral projectors.As observed in the previous article by the authors [10], it is a consequence of the ℓ 2 decoupling bound of Bourgain-Demeter.The statement is as follows.We now turn to the proof of Theorem 1.2. Proof.
Step 1: Allowing more general cutoff functions.Define the spectral projector where ζ is a Schwartz function such that ζ(0) > 0 and ζ is compactly supported.We claim that it suffices to prove Theorem 1.2 for the spectral projector P ′ λ,δ instead of P ′ λ,δ .Indeed, assume that P ′ λ,δ enjoys the bound in this theorem.Since there exists c > 0 such that ζ(x) ≥ c1 [−c,c](x) , the desired bound follows 2 for the operator . This implies in turn this bound for the operator , for a constant a > 0. Finally, this implies the desired bound for P λ,δ since |χ(x)| can be bounded by a finite sum of translates of 1 [−a,a] .
We now claim that P ′ λ,1 enjoys the Sogge bounds (1.3), just like P λ,1 .This follows from writing and bounding The rapid decay of ζ implies that |c n | n −N for any N , while 1 [λ+(n−1)δ,λ+nδ] enjoys the Sogge bounds.Thus, it is not hard to sum the above series, and deduce that P ′ λ,1 also enjoys the Sogge bounds.
By a similar argument, it can be shown that Theorem (4.1) applies to P ′ λ,1 .
Step 2: splitting the spectral projector.Writing the function x → ζ Q(x)+λ 2 λδ as a Fourier transform, the operator P ′ λ,δ becomes (4.1) with the kernel here, we choose N to be a power of 2 in the range [2λ, 4λ].The basic idea is to split the integral giving P λ,δ into two pieces, |t| < λ −1 and |t| > λ −1 .The former corresponds to an operator of the type δP ′ λ,1 , for which bounds are well-known: this corresponds to the classical Sogge theorem.The latter can be thought of as an error term, it will be bounded by interpolation between p = p ST and p = ∞, and it is for this term that genericity is used.
Turning to the implementation of this plan, we write Step 3: Bounding the term corresponding to small t.Observe that P small λ,δ can be written δP ′′ λ,1 , where P ′′ λ,1 is a variation on P ′ λ,1 ; this can be compared to the definition of P small λ,δ and (4.1).We saw in Step 1 that P ′ λ,1 enjoys the Sogge bounds, and this remains true for P ′′ λ,1 .Furthermore, by a classical T T * argument, the operator norm of the spectral projector L p ′ → L p is the square of the operator norm of the spectral projector L 2 → L p (once again, up to redefining the cutoff function χ).
Therefore, it enjoys the bound (4.2) Step 4: Bounding the term corresponding to large t.In order to bound this term, we will interpolate between • The case p = p ST : in this case, we resort to Theorem 4.1.We saw in Step 1 that it applies to P ′ λ,δ , and, by the same argument, it applies to P ′′ λ,δ .This gives • The case p = ∞: in this case, we resort to Lemma 3.4 (generic rectangular tori) and Lemma 3.6 (generic tori).In order for these lemmas to apply, we add a further requirement on ζ, namely that its Fourier transform be 1 in a neighbourhood of zero.Then, for almost any choice of (β ij ), Interpolating between these two bounds gives for almost any choice of (β ij ) Step 5: conclusion.Finally, combining (4.2) and (4.3), and using that P λ,δ (by the classical T T * argument) gives from which the desired result follows.

Some linear algebra
In this section we assemble technical tools to attack Problem 1.4.Recall that the goal is to count the number of matrices where the m ij are integers in given dyadic intervals and the maximal subdeterminants of P also lie in some specified dyadic intervals.The idea is to add the columns one by one, so that we count the number of possible (m 11 , . . ., m d1 ), and for each possibility we count the number of (m 12 , . . ., m d2 ), and so on.The main goal of this section is Lemma 5.7, which can be understood as an estimate for the measure of the real vectors (m 1k , . . ., m dk ) which are within a distance O(1) of a vector satisfying the required conditions, given the previous columns.
In this and the next section we will often use the notation k (M ) defined in section 2. 5.1.Singular values and largest subdeterminants.We begin with a number of general statements about the size of the subdeterminants of a p × q matrix, and their relation to the singular value decomposition, a type of canonical form for matrices.Throughout this subsection, implicit constants in and ∼ notation may depend on p and q.Lemma 5.1 (Singular value decomposition).Let M ∈ R p×q and let m = min(p, q).Then there are U ∈ O(p), V ∈ O(q) and (uniquely defined) singular values σ and where 0 is a matrix of zeroes (possibly empty).and M is a matrix in R p×q , (i) Proof.The statements (i) and (ii) are symmetric, so that we will only focus on (i).Finally, it follows from the uniqueness of the (σ i ) in (5.1) that σ k (U M ) = σ k (M ).
Corollary (Relation between the D k and σ k ).If k ≤ min(p, q), the singular values and the maximal subdeterminants are such that where we use the convention that 0 −1 0 = 0, or equivalently Proof.By lemmas 5.1 and 5.2, it suffices to prove these formulas for a rectangular diagonal matrix; but then they are obvious.
Lemma 5.3.Given a matrix M ∈ R p×q , we can change the order of its columns so that for each k ≤ ℓ ≤ min(p, q), (5.2) We claim first that it suffices to prove the result for the matrix U M , where U is orthogonal.Indeed, denoting M (ℓ) for the restriction of M to its first ℓ columns, this implies, in combination with Lemma 5.2, which is the desired result.For the remainder of the proof, we write for simplicity σ i = σ i (M ).
We can choose U as in Lemma 5.1, in which case, assuming for instance p ≥ q, it suffices to deal with the case M = (σ 1 L 1 , . . .σ q L q , 0, . . .0) T , where L i • L j = δ ij .The 0 entries are irrelevant, so we can assume that We now claim that, after permuting the columns of M , it can be ensured that, for any k, the top left square matrix of dimension k × k has nearly maximal subdeterminant: The construction of the matrix permutation is iterative and proceeds as follows: expanding the determinant of M with respect to the last row, we see that where M {q,i} is the matrix obtained from M by removing the q-th row and the i-th column.Since |M q,i | ≤ σ q and det M {q,i} σ 1 . . .σ q−1 for all i, we can find i 0 such that |M q,i 0 | ∼ σ q , and det M {q,i 0 } ∼ σ 1 . . .σ q−1 .Exchanging the columns i 0 and q, the resulting matrix satisfies (5.3) for k = q − 1.
We now consider the matrix N = (M ij ) 1≤i,j≤q−1 , which was denoted M {q,i 0 } before columns were permuted.It is such that entries in the last row are ≤ σ q−1 , and subdeterminants of size q − 2 are bounded by σ 1 . . .σ q−2 .Therefore, the same argument as above can be applied, and it proves (5.3) for k = q − 2.An obvious induction leads to the desired statement.5.2.Describing some convex bodies.We can use the subdeterminants studied above to describe certain convex bodies.Our first result concerns the measure of a neighbourhood of a convex hull.
There remains to evaluate D d (M ); owing to the specific structure of M , We can also describe a subset of a convex hull cut out by linear inequalities, showing that it is contained in a potentially smaller convex hull.Lemma 5.5.Given linearly independent v (1) , . . .v (d) ∈ (R d ) d , and Y i > 0, Z i > 0 there are w (1) , . . .w (d) ∈ (R d ) d , with (5.4) w Proof.Let Y be the matrix with columns Y i v (i) , which, without loss of generality, can be assumed to have nondecreasing norms.We claim that its singular values, τ i , are such that Indeed, by the Courant minimax principle, τ k can be characterized as follows Let Z be the matrix diag(Z 1 , . . ., Z d ), and let M = (Y −T | Z −T ) T ∈ R 2d×d .Then the set on the left-hand side of (5.5) is contained in {z : |M z| 1}.By Lemma 5.1, we can write M = U (Σ T | 0) T V , so that the set {z : |M z| 1} can now be written (up to a multiplicative constant) as W B(0, 1), with We can now define the w (i) to be the columns of W ; in order to establish the lemma, it suffices to prove the inequality (5.4).Note first that (5.7) Combining (5.6), (5.7) and (5.8), Finally, W = ZU 3 , which gives |w 5.3.Extending matrices with prescribed largest subdeterminants.We now start to describe the columns which may be added to a given p × k matrix, with a prescribed effect on its singular values.
Lemma 5.6.Let M be a p × k matrix, which admits a singular value decomposition as in (5.1).
For some fixed C > 0, let and set Then, denoting U (i) for the columns of the matrix U from the singular value decomposition of M , Proof.In the proof we allow all implicit constants in , ∼ notation to depend on C, p, k.
Step 1: p ≥ k + 1 and U = Id.Then the singular value decomposition of M is M = Σ 0 V and ) , by considering submatrices consisting of the first k rows, together with one of the last p − k lines).Furthermore, by considering submatrices consisting of a (k − 1) × (k − 1) submatrix of ΣV , one of the p − k last rows, and the last column, we have It follows that |x ′′ | σ k (M ).
(where V (i) stands for the i-th row of V ).We now prove by induction on n that |x i | ≤ σ i (M ) if i ≤ n; this assertion for n = k is the desired result.The case n = 1 being immediate, we can assume the assertion holds at rank n, and aim at proving it at rank n + 1.
The n first rows of V are orthogonal, therefore we can delete the last n − k rows and some k − n columns of V to get an n × n matrix with a determinant ∼ 1; denote this matrix V and its rows V (1) , . . ., V (n) .
Note that the n × n matrix with rows σ 1 (M ) V (1) , . . ., σ n (M ) V (n) has determinant ∼ D n (M ).We now consider the submatrix of M obtained by deleting the last n − k − 1 rows and the same columns that were deleted from V to make V .That is, x 1 . . . . . .
We further write M {i,n+1} for the matrix M with i-th row and last column removed.Expanding the determinant of M with respect to the last column, we find that .
By the induction assumption, and we saw that det M {n+1,n+1} ∼ D n (M ).Finally, the definition of S(M, R) requires that det( M ) D n+1 (M ).Combining these observations and the above equality implies that, if Step 2: general case p ≥ k + 1.Then the singular value decomposition of Setting y = U −1 x, we can write Then x ∈ S(M, R) if and only if y ∈ S(DV, R) ⊂ {y : The desired result follows for x = U y.
Step 3: the case p ≤ k.Similarly to the case p ≥ k + 1, one deals first with U = Id.Then Proceeding as in Step 1, one can deduce that |x i | τ i if 1 ≤ i ≤ p, and the desired conclusion follows as in Step 2.
We can apply the last lemma to Problem 1.4, with some technical complexity coming from the constant entries in the last row of the matrix P appearing there.In the following lemma one should think of M as the first k columns of P , and x as being a column (m 2 1(k+1) , . . ., m 2 d(k+1) , λ 2 ) T to be adjoined to the matrix M .As the m i(k+1) range over integers of size ∼ µ i , the vector T then takes values which are separated from each other by distances 1.In section 6.2 we will use this to bound the number of integral m i(k+1) by the measure of a neighbourhood of the permissible real vectors Mx.It is this measure that is estimated in (5.9).Lemma 5.7.Adopting the notation of Lemma 5.6, let µ 1 ≥ . . .≥ µ p−1 > 0 and let M be the (p − 1) × p matrix defined by M = (diag(µ −1 1 , . . ., µ −1 p−1 )|0).As in Lemma 5.6 let M be a p × k matrix, fix C > 0 and put ) for i ≤ k or i > k respectively.Then, for any A > 0, if M p1 > ǫσ 1 (M ) for some ǫ > 0, then (5.9) mes{Mx+w : where W is a (p − 1) × (p − 1) matrix with entries such that Proof.In the proof we allow all implicit constants in , ∼ notation to depend on C, p, k.Taking the difference of two vectors in the set on the right-hand side of (5.9), we see that it suffices to prove the desired statement for A = 0, and the condition In other words, it suffices to prove mes{Mx + w : Define the projector P on the first p − 1 coordinates of a vector of R p : Let U, V be matrices as in (5.1) and let U (i) be the ith column of U .Since M p1 > ǫσ 1 (M ) there is i 0 such that [M V −1 ] pi 0 ǫσ 1 (M ), that is to say By Lemma 5.6, {x ∈ S(M, R) : Since the p-th coordinate of p i=1 y i U (i) is 0, we find that where and our choice of i 0 above ensures that | U (i) | ǫ 1.Therefore, For the proof, note that we cannot have every y ii = 0 or else (y ij ) would vanish.We assume without loss of generality that there is i 0 ∈ {1, . . ., d} such that y ii = 0 iff i > i 0 .There are O(Y i 0 ) possible values of y ii for i ≤ i 0 , and once these are chosen the identity y ii = (x i , and so up to finitely many choices both of these are determined by the values of y 1i = (±x i , for which there are O(Y d−i 0 ) possibilities.We conclude that there are ǫ Y d+ǫ choices for the x (i) and hence for (y ij ).
We can now conclude that, for a suitably large constant C depending only on the cutoff function χ, and for any fixed values of the off-diagonal entries ). Applying the Borel-Cantelli lemma (Lemma A.1) proves (1.8).
We now begin the proof of Theorem 6.1.Throughout the rest of this section we write β i for β ii , put β ′ = (β 1 , . . .β d ) T , and given b, d ∈ N and M = (m ij ) 1≤i≤d, 1≤j≤b , we put 6.1.Integrating over λ and β.Our key observation is as follows.Since, for m ∈ Z, we have 1 ≤ µ∈2 N ∪{0} 1 µ≤2m≤2µ , and since χ takes non-negative values, we have for any λ, δ > 0 that and if we temporarily write the off-diagonal parts of Q(m 1j , . . ., m dj ) using the row vector then this becomes We can estimate the measure inside the last sum in (6.4) as follows.Notice first that (6. . Together with (6.4) and the fact that P (M ) does not depend on the signs of the m ij , we find , where the m ij are now non-negative since m ij ∈ [µ i /2, µ i ].Combining this with (6.5) and the Corollary to Lemma 5.2 yields (6.6) In (6.6) we may assume that (µ i = 0 =⇒ L i+1 = 0), since otherwise Z d,b ( µ, L) would be zero (there are no such M ).In particular allowing µ i to be zero is the same as allowing the dimension d to drop, in the sense that max

6.2.
Counting matrices with prescribed subdeterminants.We want to estimate the righthand side of (6.6), under the assumption that µ i = 0 for any i, or in other words µ i ∈ 2 N .Our first object is to estimate from above the number of matrices M counted by the function Z d,b ( µ, L) from (6.7).By Lemma 5.3, it suffices to l count those M satisfying an additional condition That is the vectors m (j) are the columns of M .Meanwhile the vectors n (j) are the columns of P (M ) with the last element dropped and the others rescaled to that n (j) belongs to the set S defined by whose elements are separated by gaps of size ∼ 1.If M is counted in the right-hand side of (6.7), then the vector n (1) can be chosen arbitrarily from S; there are d i=1 µ i choices.Suppose now that the first k columns of P (M ) are given, and they satisfy We want to select m (k+1) , or equivalently n (k+1) .We can first use that S is 1-separated to replace our counting problem by a volume estimate: letting Remark 6.2.This volume bound is not necessarily optimal.To take just one simple example, if λ 2 0 is an integer then D (ℓ) ℓ (P (M )) is an integer.Thus, in the case when 0 hold and the set N k+1 is empty.We apply Lemma 5.7 with We compute that We now need to distinguish two cases.
Recall now that this is a bound for the number of choices for m (k+1) , given m (1) , . . ., m (k) , and that there are µ 1 • • • µ d choices for m (1) .Recall also that our object is to estimate that part of the right-hand side of (6.6) for which every µ i is nonzero.The bound we have proved is (6.8) max where σ k is a bijection from I k to a subset of {1, . . ., d} and with the understanding that I might be empty and I = 1 if I = ∅.
6.3.The maximization procedure.Our aim is now to find the values of L i , µ i , I k , σ k for which the maximum on the right-hand side of (6.8) is attained.
Step 1: Maximizing in (L i ) with the I k 's and µ i 's held fixed.We start with the dependence on (L i ), and we relax first the condition that they be ordered; we will simply assume that 0 ≤ L i ≤ λ 2 0 for each i = 1, . . ., min(b, d + 1).Next, we claim that the products Therefore, F will be larger, or equal, if we increase the value of max(L σ k (i)+1 , L k ) to µ 2 i 1 , which has the effect of cancelling the undesirable term.
After this manipulation, the parentheses we mentioned have been cancelled, and the value of some of the L i 's has been fixed to µ 2 f (i) for some function f .The remaining L i contribute 1 L i >δλ 0 δλ 0 L i , and they might be constrained by inequalities of the type L i > µ 2 j .Therefore, F will be maximal if they take the value δλ 0 , or µ 2 f (i) for some function f .
Step 2: Maximizing in (µ i ).The result of the maximization in (L i ) is that we can assume that each L i takes the value either µ 2 f (i) , or δλ 0 , that µ 2 i ≤ max(L σ k (i)+1 , L k+1 ) if i ∈ I k , with the convention that L k = 0 for k ≥ d + 2, and that the function to maximize is (6.9) λ 0 i=1,...,min(b,d+1) We now claim that, at the maximum, the (µ i ) either take the value 1 or λ 0 .To prove this claim, assume that, at the maximum, the µ i take a number n of distinct values 1 ≤ a 1 < • • • < a n ≤ λ 0 .Replacing L i by µ 2 f (i) in the above expression, it takes the form λ 0 (λ 0 δ) α 0 n i=1 a α i i , where α i ∈ Z.
If α i > 0 and a i < λ 0 , then this expression will increase if the value of a i is increased until a i+1 or λ 0 ; and similarly, if α i < 0 and a i > 1, it will decrease if the value of a i is decreased until a i−1 or 1.This contradicts the maximality of (µ i ) unless a i only takes the values λ 0 or 1 for α i = 0.There remains the case where α i = 0, but then µ i can be assigned the value λ 0 or 1 indifferently.
Step 3: Maximizing in (I k ) and (σ k ).We showed that the maximum of F is less than the maximum of (6.9), under the constraint that µ 2 i ≤ max(L σ k (i)+1 , L k+1 ) if i ∈ I k (with the convention that L k = 0 for k ≥ d + 2); and under the further constraint that L i can only take the values δλ 0 , 1, λ 0 , and µ i can only take the values 1, λ 0 .
There are now two cases to consider: • If k ≤ min(b − 1, d) and L k+1 = λ 2 0 , then the optimal choice for I k is {1, . . ., d}. • Otherwise, I k should have the same cardinal as the set of L i , i ≥ 2, equal to λ 2 0 , and σ k + 1 should map I k to this set.µ i .
Notice that we are assuming again that the (µ i ) and (L i ) are ordered; a moment of reflection shows that this is possible since the permutation (σ k ) can be freely chosen.This expression is visibly nondecreasing in (µ i ), so we might as well take all µ i to be λ 0 .
In order to evaluate the resulting expression, we need to know the number of L k equal to, respectively, δλ 0 , 1, λ 0 ; this is also the information needed to determine I k ; therefore, we define the We notice first that b 1 can be taken to be zero.Second, there remains to dispose of the assumption that all µ i are non-zero, which was made at the beginning of Subsection 6.2.By the comments just prior to the start of Subsection 6.2, this equivalent to reducing the dimension d.But some thought shows that the above expression is increasing with d, so that allowing for smaller d is harmless.Overall, the final bound we find is max This will be ≥ 1 provided that δ ≥ λ min(b−d−1,d+1−b) , and so for such δ the maximum in (6.10) is reached for b 2 = min(b, d + 1).We will therefore impose the condition δ ≥ λ min(b−d−1,d+1−b) for convenience rather than because we believe it to be optimal.This yields

D
k (P (M )) ∼ D (k) k (P (M )) for all ≤ k ≤ min(b, d + 1), since permuting the columns of these recovers all the matrices in Z d,b ( µ, L) For j = 1, . . ., b define the vectors m (j) and n (j) ∈ R d by
Boundedness of spectral projectors on Riemannian manifolds.Given a Riemannian manifold M with Laplace-Beltrami operator ∆, and given some λ ≥ 1 and 0 < δ < 1, let If k ≤ d then applying Lemma 5.7 gives that