Diophantine approximation on manifolds and the distribution of rational points: contributions to the convergence theory

In this paper we develop the convergence theory of simultaneous, inhomogeneous Diophantine approximation on manifolds. A consequence of our main result is that if the manifold $M \subset \mathbb{R}^n$ is of dimension strictly greater than $(n+1)/2$ and satisfies a natural non-degeneracy condition, then $M$ is of Khintchine type for convergence. The key lies in obtaining essentially the best possible upper bound regarding the distribution of rational points near manifolds.


Introduction and statement of results
1.1. The setup. Throughout, we suppose that m ≤ d, n = m+d and that f = (f 1 , . . . , f m ) is defined on U = [0, 1] d . Suppose further that ∂f /∂α i and ∂ 2 f /∂α i ∂α j exist and are continuous on U , and that there is an η > 0 such that for all α ∈ U (1. 1) det Throughout R + = [0, +∞) is the set of non-negative real numbers. Let ψ : R + → R + be a function such that ψ(r) → 0 as r → ∞ and θ = (λ, γ) ∈ R d × R m . Now for a fixed q ∈ N, . . , f m (α)) ∈ R n : α = (α 1 , . . . , α d ) ∈ U } embedded in R n . Recall that by the Implicit Function Theorem any smooth manifold M can be locally defined in this manner; i.e. with a Monge parametrisation. The upshot is that, A(q, ψ, θ) counts the number of shifted rational points a 1 +λ 1 q , . . . , a d +λ d q , b 1 +γ 1 q , . . . , bm+γm q ∈ R n that lie (up to an absolute constant) within the ψ(q)/q-neighbourhood of M f . Before stating our counting results it is worthwhile to compare condition (1.1) imposed on the Jacobian of f with that of non-degeneracy as defined by Kleinbock and Margulis in their pioneering work [11]. In this paper they prove the Baker-Sprindžuk 'extremality' conjecture in the theory of Diophantine approximation on manifolds.
The above map f : U → R m : α → f (α) = (f 1 (α), . . . , f m (α)) is said to be l-nondegenerate at α ∈ U if there exists some integer l ≥ 2 such that f is l times continuously differentiable on some sufficiently small ball centred at α and the partial derivatives of f at α of orders 2 to l span R m . The map f is called non-degenerate if it is l-non-degenerate at almost every (in terms of d-dimensional Lebesgue measure) point in U ; in turn the manifold M f is also said to be non-degenerate. Non-degenerate manifolds are smooth sub-manifolds of R n which are sufficiently curved so as to deviate from any hyperplane at a polynomial rate see [1,Lemma 1(c)]. As is well known, see [11, p. 341], any real connected analytic manifold not contained in any hyperplane of R n is non-degenerate.
It follows from the definition of l-non-degeneracy, that condition (1.1) imposed on f implies that f is 2-non-degenerate at every point. Although (1.1) is fairly generic, the converse is not always true even if we allow rotations of the coordinate system. The submanifold (x, y, z 1 , . . . , z k , x 2 , xy, y 2 ) of R k+5 provides a counterexample 1 .

1.2.
Results on counting rational points. Throughout, the Vinogradov symbols ≪ and ≫ will be used to indicate an inequality with an unspecified positive multiplicative constant. If a ≪ b and a ≫ b, we write a ≍ b and say that the two quantities a and b are comparable. Throughout the paper the constants will only depend on the dimensions n and d and the map f .
Observe that for q sufficiently large so that ψ(q) ≤ 1/2 , we have that where as usual x := max 1≤i≤m x i for any x ∈ R m . In particular, when 0 < ψ(q) ≤ 1/2, the obvious heuristic argument leads us to the following estimate: We establish the following upper bound result.
where the implied constant is independent of q, θ and ψ but may depend on f .
The following is a straightforward consequence of the theorem. It states that the upper bound (1.6) coincides with the heuristic estimate if ψ(q) is not too small.
Then for integers q ≥ 2 we have that 1.3. Results on metric Diophantine approximation. Given a function ψ : R + → R + and a point θ = (θ 1 , . . . , θ n ) ∈ R n , let S n (ψ, θ) denote the set of y = (y 1 , . . . , y n ) ∈ R n for which there exists infinitely many q ∈ N such that In the case that the inhomogeneous part θ is the origin, the corresponding set S n (ψ) := S n (ψ, 0) is the usual homogeneous set of simultaneously ψ-approximable points in R n . In the case ψ is ψ τ : r → r −τ with τ > 0, let us write S n (τ, θ) for S n (ψ, θ) and S n (τ ) for S n (τ, 0). Note that in view of Dirichlet's theorem (n-dimensional simultaneous version), S n (τ ) = R n for any τ ≤ 1/n.
In the general discussion above we have not made any assumption on ψ regarding monotonicity. Thus the integer support of ψ need not be N. Throughout, N ⊂ N will denote the integer support of ψ. That is the set of q ∈ N such that ψ(q) > 0. Regarding the set S n (ψ, θ), measure theoretically, this is equivalent to saying that we are only interested in integers q lying in some given set N such as the set of primes or squares or powers of two. The theory of restricted Diophantine approximation in R n is both topical and well developed for certain sets N of number theoretic interest -we refer the reader to [10,Chp 6] and [3, §12.5] for further details. However, the theory of restricted Diophantine approximation on manifolds is not so well developed.
Armed with Corollary 1, we are able to establish the following convergent statement for the s-dimensional Hausdorff measure H s of M f ∩ S n (ψ, θ). Note that if s > d = dim M f , then H s (M f ∩ S n (ψ, θ)) = 0 irrespective of ψ. This follows immediately from the definition of Hausdorff dimension and that fact that Theorem 2. Let θ ∈ R n and ψ : R + → R + be a function such that ψ(r) → 0 as r → ∞ and where as N = {q ∈ N : ψ(q) > 0}. Let 0 < s ≤ d and f : U → R m satisfy the following condition (1.9) H s α ∈ U : the l.h.s. of (1.1) = 0 = 0. Then Remark 1. Recall, that in view of the discussion in §1.1 the condition imposed on f in the above theorem and its corollaries below are equivalent to saying that the manifold is 2-nondegenerate everywhere except on a set of Hausdorff s-measure zero.
Now we consider two special cases of Theorem 2. First suppose the integer support of ψ is along a lacunary sequence. In particular, consider the concrete situation that N = {2 t : t ∈ N}. The following statement is valid for any n = d + m and to the best of our knowledge is first result of its type even within the setup of planar curves (d = m = 1).

Corollary 2.
Let θ ∈ R n and ψ : R + → R + be a function such that ψ(r) → 0 as r → ∞ and N = {2 t : t ∈ N}. Let d − n 2(m+1) < s ≤ d and assume that f : U → R m satisfies (1.9). Then Proof. Consider the auxiliary functioñ where C > 0 is a sufficiently large constant. Then as is easily verified using the convergence sum condition of Corollary 2 ∞ t=1 2 −tψ (2 t ) s+m 2 tn < ∞ and therefore, by Theorem 2, we have that H s M f ∩ S n (ψ, θ) = 0. Trivially, we have that S n (ψ, θ) ⊂ S n (ψ, θ) and then the required statement follows on using the monotonicity of H s .
Let us now consider Theorem 2 under the assumption that ψ is monotonic. Then, without loss of generality, we can assume that N = N since otherwise ψ(q) = 0 for all sufficiently large q and so S n (ψ, θ) is the empty set and there is nothing to prove. Furthermore, we can assume that ψ(q) ≪ q −1/n for all q ∈ N since otherwise the s-volume sum appearing in the theorem is divergent for s ≤ d. This is in line with the fact that if ψ(q) ≥ q −1/n for all sufficiently large q, then by Dirichlet's theorem we have that The upshot is that within the context of Theorem 2, for monotonic ψ we can assume that This forces d > (n + 1)/2.

Corollary 3.
Let θ ∈ R n and ψ : R + → R + be a monotonic function such that ψ(r) → 0 as r → ∞. Let d > n+1 2 and s 0 := dm m+1 + n+1 2(m+1) < s ≤ d and assume that f : U → R m satisfies (1.9). Then The proof is similar to that of Corollary 2. Note that (1.9) is always satisfied if Also note that the condition d > (n + 1)/2 guarantees that s 0 < d. However, it does mean that the corollary is not applicable when n = 3 or n = 2. The fact that is not applicable when n = 2 is not a concern -see Remark 2 immediately below.
Remark 2. It is conjectured that the conclusion of Corollary 3 is valid for any non-degenerate manifold (i.e. d ≥ 1) and dm (m+1) < s ≤ d -see for example [2, §8]. For planar curves (d = m = 1), this is known to be true [5,14]. To the best of our knowledge, beyond planar curves, the corollary represents the first significant contribution in favour of the conjecture.

Remark 3. Corollary 3 together with the definition of Hausdorff dimension implies that if
Lebesgue measure on M f . In other words, it proves that the 2-non-degenerate submanifold M f of R n with dimension strictly greater than (n + 1)/2 is of Khintchine-type for convergence -see [4]. Apart from the planar curve results referred to in Remark 2, the current state of the convergent Khintchine theory is somewhat ad-hoc. Either a specific manifold or a special class of manifolds satisfying various constraints is studied. For example it has been shown that (i) manifolds which are a topological product of at least four non-degenerate planar curves are Khintchine type for convergence [7] as are (ii) the so called 2-convex manifolds of dimension d ≥ 2 [9] and (iii) straight lines through the origin satisfying a natural Diophantine condition [12].
Remark 5. In view of the conjecture mentioned above in Remark 2, we expect (1.10) to remain valid for any non-degenerate manifold without any restriction on its dimension. Note that it is relatively straightforward to establish that this is indeed the case for almost all θ. Moreover, we do not need to assume that ψ is monotonic or even that M f is non-degenerate. In other words, for any C 1 submanifold 2 M f of R n and ψ : R + → R + , we have that (1.10) is valid for almost all θ ∈ R n . This is an immediate consequence of the following even more general 'doubly metric' result.
Proposition 1. Let f : U → R n be any continuous map. Given ψ : Proof. The proposition is pretty much a direct consequence of Fubini's theorem. Without loss of generality, we can assume that θ is restricted to the unit cube [0, 1] n . For q ∈ N, let 2 By a C 1 submanifold we mean an immersed manifold into R n by a C 1 map, that is the image of a C 1 map Notice that and that by Fubini's theorem and the Borel-Cantelli lemma implies the desired measure zero statement.
1.4. Restricting to hypersurfaces. As already mentioned, the condition d > (n + 1)/2 means that Corollary 3 is not applicable when n = 3. We now attempt to rectify this. In the case m = 1, so that the manifold M f associated with f is a hypersurface, we can do better than Theorem 1 if we assume that M f is genuinely curved. More precisely, in place of (1.1) we suppose that there is an η > 0 such that for all α ∈ U where for brevity we have written f for f 1 . It is not too difficult to see that this condition imposed on the determinant (Hessian) is valid for spheres but not for cylinders with a flat base. We will refer to the hypersurface M f with f satisfying (1.12) as genuinely curved.
Throughout the rest of this section we will assume that m = 1 and so d = n − 1.
Theorem 3. Suppose that f : U → R satisfies (1.12) and θ ∈ R n . Suppose that 0 < ψ(q) ≤ 1/2. Then where the implied constant is independent of q, θ and ψ but may depend on f .
A simple consequence of this theorem is the following analogue of Corollary 1.
Corollary 4. Suppose that f : U → R satisfies (1.12) and θ ∈ R n . Suppose that Then for integers q ≥ 2 we have that It is easily seen that Theorem 1 with m = 1 and Theorem 3 coincide when n = 2 but for n ≥ 3 the second term on the r.h.s. in (1.13) is smaller than the corresponding term in (1.6).
In particular, and so Corollary 4 is stronger than Corollary 1 for f satisfying (1.12). Corollary 4 enables us to obtain the analogue of Theorem 2 for genuinely curved hypersurfaces in which the condition that ψ(q) ≫ q −1/(2m+1) (log q) 2/(2m+1) for q ∈ N is replaced by ψ(q) ≫ q −d/(2+d) (log q) 2d/(2+d) for q ∈ N . In turn for monotonic functions we have the following statement. It represents a strengthening of Corollary 3 in the case of genuinely curved hypersurfaces and is valid when n = 3.
Corollary 5. Suppose that f : U → R and θ ∈ R n . Let ψ : R + → R + be a monotonic function such that ψ(r) → 0 as r → ∞. Let n ≥ 3 and n−1 2 + n+1 2n < s ≤ n − 1 and assume that H s α ∈ U : the l.h.s. of (1.12) = 0 = 0. Then The conjectured lower bound for s above is (n−1)/2 -see Remark 2 preceding the statement of Corollary 3. The proof of the above corollary is similar to that of Corollary 2.
1.5. Further remarks and other developments. The upper bound results of §1.2 for the counting function A(q, ψ, θ) are at the heart of establishing the convergence results of §1.3. We emphasize that A(q, ψ, θ) is defined for a fixed q and that Theorem 1 provides an upper bound for this function for any q sufficiently large. It is this fact, that enables us to obtain convergent results such as Theorem 2 without assuming that ψ is monotonic. While statements without monotonicity are desirable, considering counting functions for a fixed q does prevent us from taking advantage of any potential averaging over q. More precisely, for Q > 1 consider the counting function If ψ is monotonic, then ψ(q) ≤ ψ(Q) for Q < q ≤ 2Q and the obvious heuristic 'volume' argument leads us to the following estimate: Clearly, the upper bound (1.7) for A(q, ψ, θ) as obtained in Corollary 1 implies (1.16). The converse is unlikely to be true. However, for monotonic ψ establishing (1.16) suffices to prove convergence results such as Corollary 3. Indeed, the fact that we have a complete convergence theory for planar curves (see Remark 2 in §1.3) relies on the fact that we are able to establish (1.16) with m = 1 = d. Note that the counting result obtained in this paper for A(q, ψ, θ) is not strong enough to imply any sort of convergent Khintchine type result for planar curves with ψ monotonic. Furthermore, it is worth pointing out that averaging over q when considering N (Q, ψ, θ) also has the potential to weaken the lower bound condition (1.8) on ψ appearing in Theorem 2. This in turn would increase the range of s within Corollaries 3 and 5.
Regarding lower bounds for the counting function N (Q, ψ, θ), if ψ is monotonic, then ψ(q) ≥ ψ(Q) for 1 2 Q < q ≤ Q and the heuristic 'volume' argument leads us to the following estimate: In the homogeneous case (i.e. when θ = 0), the lower bound given by (1.17) is established in [2] for any analytic non-degenerate manifold M embedded in R n and ψ satisfying lim q→∞ qψ(q) m = ∞. When M is a curve, the condition on ψ can be weakened to lim q→∞ qψ(q) (2n−1)/3 = ∞. Moreover, it is shown in [2] that the rational points a/q associated with N ( 1 2 Q, ψ, 0) are 'ubiquitously' distributed for analytic non-degenerate manifolds. This together with the lower bound estimate is very much at the heart of the divergent Khintchine type results obtained in [2] for analytic non-degenerate manifolds. In a forthcoming paper [6], we establish the lower bound estimate (1.17) and show that shifted rational points a+λ q associated with N ( 1 2 Q, ψ, θ) are 'ubiquitously' distributed for any C n+1 non-degenerate curve in R n and arbitrary θ. As a consequence, we obtain a divergent Khintchine type theorem for Hausdorff measures. More specifically, let f = (f 1 , . . . , f n−1 ) : [0, 1] → R n−1 be a C n+1 function such that for almost all α ∈ [0, 1] Let 1 2 < s ≤ 1, θ ∈ R n and ψ : R + → R + be a monotonic function such that ψ(r) → 0 as r → ∞. It is established in [6] that In view of the conditions imposed on f above, the associated manifold M f is by definition a C n+1 non-degenerate curve in R n . When s is strictly less than one, non-degeneracy can be replaced by the condition that (1.18) is satisfied for at least one point α ∈ [0, 1]. In other words, all that is required is that there exists at least one point on the curve that is non-degenerate. Using fibering techniques, it is also shown in [6] that the above statement for non-degenerate curve in R n can be readily extended to accommodate a large class of non-degenerate manifolds beyond the analytic ones considered in [2].

Preliminaries to the proofs of Theorems 1 and 3
To establish Theorems 1 and 3 we adapt an argument of Sprindžuk [13,Chp2 §6]. In our view the adaptation is non-trivial.
Otherwise, the error term associated with (1.6) is, up to a multiplicative constant, larger than the trivial bound A(q, ψ, θ) ≤ (q + 1) d and there is nothing to prove. Now define By the mean value theorem for second derivatives, when a ∈ A(q, ψ, θ, u), Here the error term is where C 1 depends at most on d and the size of the second derivatives. Now choose Thus, for a = ru + v with a ∈ A(q, ψ, θ, u) we have By the definition of H, we have that On using the fact that e(x 1 + · · · + x ℓ ) = e(x 1 ) · · · e(x ℓ ) and |e(x)| = 1 for any real numbers x, x 1 , . . . , x ℓ , we find that Therefore, by (2.24), it follows that Since R = [0, r) ∩ Z, for any given ρ ∈ R we have that v∈R e(vρ) ≤ r and also that v∈R where the implied constant is absolute. Hence, on taking ρ = m j=1 h j This together with (2.25), implies that Moreover where the left hand side inequality follows from the definitions of r and H -see (2.20) and (2.22). Hence and furthermore, by considering their product over i, we get that Since the measure of I 1 × · · · × I d is ≍ 1, integrating the above inequality over β ∈ I 1 × · · · × I d gives that Now recall that the rectangles I 1 × · · · × I d depend on the choice of u. Note that their union taken over integer points u ∈ S d , where S := [0, s], is exactly S d . Furthermore, different rectangles can only intersect on the boundary. Hence summing the above displayed inequality over all integer points u ∈ S d gives Now combining this together with (2.26) we obtain that Now finally observe that

The proof of Theorem 1
With reference to §2, by (2.27) Since (1.1) holds we may make the change of variables Thus The contribution from h = 0 is is constant with respect to ω m+1 , . . . , ω d . Hence, by Fubini's theorem and the fact that If h = 0, then assuming for example that h 1 = 0 and using Fubini's theorem again we get that Hence, by the above inequalities and (3.2), the contribution from the h = 0 terms within (3.1) is estimated by In view of (2.28), it follows that Given the definitions of H and r this gives (1.6) and thereby completes the proof of the theorem.

The proof of Theorem 3
Recall that within Theorem 3 we have that m = 1 and d = n − 1. Hence, with reference to §2, (2.27) becomes where f = f : U → R. Since (1.12) holds we may make the change of variables and The contribution from h = 0 is and the contribution from the remaining terms is In view of (2.28), it follows that Given the definitions of H and r this gives (1.13) and thereby completes the proof of the theorem.

Proof of Theorem 2
Step 1.
As mentioned in §1, in view of the Implicit Function Theorem, we can assume without loss of generality that the manifold M f is of the Monge form (1.3). Note that, since U is compact and f is C 1 , this implies via the Mean Value Theorem that f = (f 1 , . . . , f m ) is bi-Lipschitz and so there exists a constant c 1 ≥ 1 such that Let Ω f n (ψ, θ) denote the projection of M f ∩ S n (ψ, θ) onto U ; that is Ω f n (ψ, θ) := {α ∈ U : (α, f (α)) ∈ S n (ψ, θ)} . Explicitly, given θ = (λ, γ) ∈ R d × R m , the set Ω f n (ψ, θ) consists of points α ∈ U such that the system of inequalities is satisfied for infinitely many (q, a, b) ∈ N × Z d × Z m . Furthermore, there is no loss of generality in assuming that a+λ q ∈ U for solutions of (5.2). In view of (5.1), the sets Ω f n (ψ, θ) and M f ∩ S n (ψ, θ) are related by a bi-Lipschitz map and therefore H s (M f ∩ S n (ψ, θ)) = 0 ⇐⇒ H s (Ω f n (ψ, θ)) = 0 . Hence, it suffices to show that (5.3) H s (Ω f n (ψ, θ)) = 0 .
Notice that the set B = {α ∈ U : l.h.s. of (1.1) = 0} is closed and therefore G = U \ B can be written as a countable union of closed rectangles U i on which f satisfies (1.1). The constant η associated with (1.1) depends on the particular choice of U i . For the moment, assume that H s (Ω f n (ψ, θ) ∩ U i ) = 0 for any i ∈ N. On using the fact that H s (B) = 0, we have that H s Ω f n (ψ, θ) ∩ U i = 0 and this establishes (5.3). Thus, without loss of generality, and for the sake of clarity we assume that f satisfies (1.1) on U .
Step 3. For a point p+θ q ∈ R n with p = (a, b) ∈ Z d × Z m , let σ p+θ q denote the set of α ∈ U satisfying (5.2). Trivially, where the implied constant depends on n only.
By definition, the right hand side is simply the counting function A(q, c 2 ψ, θ). Thus, by Corollary 1, for q ∈ N sufficiently large we have that (5.5) # p ∈ Z n : σ( p+θ q ) = ∅ ≪ ψ(q) m q d .
This completes the proof of Theorem 2.