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Some manifolds of Khinchin type for convergence

Recently, Beresnevich, Vaughan, Velani, and Zorin (arXiv: 1506.09049) gave some sufficient conditions for a manifold to be of Khinchin type for convergence. We show that their techniques can be used in a more optimal way to yield stronger results. In the process we also improve a theorem of Dodson, Rynne, and Vickers ('89).


Khinchin-type results
Fix n ∈ N. It is an easy consequence of the Borel-Cantelli lemma that if ψ : N → [0, ∞) is a function such that the series converges, then for all θ ∈ R n , the set S(ψ, θ) def = {x ∈ R n : ∃ ∞ q ∈ N qx − θ < ψ(q)} is of Lebesgue measure zero. Here · denotes distance to the nearest integer vector, measured using the max norm, which we denote by | · |, and ∃ ∞ means "there exist infinitely many". The preceding result is known as the convergence case of Khinchin's theorem. A manifold M ⊂ R n is said to be of Khinchin type for convergence if its typical points behave like the typical points of Lebesgue measure with respect to this theorem. More precisely, let us say that M is of strong (resp. weak) Khinchin type for convergence if for every function (resp. monotonic function) ψ satisfying (1.1) and for all (1) The use of the adjectives "strong" and "weak" in this context is new. In the literature, the phrase "Khinchin type for convergence" usually means "weak Khinchin type for convergence". (2) Although the authors of [2] only prove the special case K = [0, 1] d , their arguments work just as well for the general case. (3) Although the statements of [2,Corollaries 3 and 5] only yield that M is of weak Khinchin type for convergence, the proofs actually show that M is of strong Khinchin type for convergence, since the assumption that ψ is monotonic is not used anywhere in the proofs. Theorem 1.2 ([6, Theorem 1.3]). Fix d, m ∈ N, let K ⊂ R d be a closed rectangle, let f : K → R m be a function of class C 2 , and let M ⊂ R d+m be as in (1.2). If m ≤ d 2 , and for Lebesgue-a.e. α ∈ K, (1.5) for all t ∈ R m \ {0}, the matrix t · f (α)[e i , e j ] 1≤i,j≤d has at least two nonzero eigenvalues that share the same sign, then M is of strong Khinchin type for convergence.
Here the assumption m ≤ d 2 does not appear in [6], but we have added it because the hypothesis (1.5) is not satisfiable when m > d 2 (Theorem 5.1 (v)). It appears to be a difficult problem to determine precisely for which values m ≤ d 2 the hypothesis is satisfiable; cf. the discussions in [5, §2] and in Section 5. We have made progress on this problem by showing that the hypothesis is satisfied generically whenever m ≤ d−1 2 (Theorem 5.1 (vii)).
The main goal of this paper is to generalize Theorems 1.1 and 1.2 simultaneously, yielding a new theorem more powerful than both of them. In the following theorem, Case (1) is a generalization of Case (1) of Theorem 1.1, and Case (2) is a generalization of Case (2) of Theorem 1.1 and also of Theorem 1.2: Theorem 1.3. Fix d, m ∈ N, let K ⊂ R d be a closed rectangle, let f : K → R m be a function of class C 2 , and let the manifold M ⊂ R d+m be given by (1.2). If we are in either of the following scenarios: (1) m < d − 1, and for Lebesgue-a.e. α ∈ K, (2) m < d+1 2 , and for Lebesgue-a.e. α ∈ K, we have then M is of strong Khinchin type for convergence.
We now show that this theorem is in fact a generalization of Theorems is satisfiable for all m < d+1 2 ; we refer to Section 5 for details, specifically Theorem 5.1 (ii). In Section 5, we also show that to enforce that "almost all" functions f satisfy (1.7), the stronger inequality m ≤ d 2 is needed (Theorem 5.1 (iii) and (iv)). Remark 1.6. Theorems 1.1, 1.2, and 1.3 appear to be the only known results regarding manifolds of strong Khinchin type for convergence (for simultaneous approximation). However, there are some results regarding manifolds of weak Khinchin type for convergence; for example, it was proven in [8, Theorem 5] (see also [1]) that nondegenerate planar curves are of weak Khinchin type for convergence. The results of this paper do not apply to planar curves, since the parameters d = m = 1 do not satisfy the dimension constraints. It remains an open question whether nondegenerate planar curves (e.g. the standard parabola {(x, x 2 ) : x ∈ R}) are of strong Khinchin type for convergence.
Outline of the paper. In Sections 2 and 3 we continue to state our main results, each time reducing the main result of the previous section to the main result of the current section. Then in Section 4 we prove the main result of Section 3, and thus by implication all of the main results, using technical tools from [2]. In Section 5 we discuss the significance of the hypothesis (1.7), answering the question of how commonly it is satisfied.
In what follows, we do not give an exhaustive comparison of our results with the corresponding results in [2] and [6]; comparing Theorems 1.1 and 1.2 vs. Theorem 1.3 illustrates the main differences. However, we do make the observation that [6, (5.1)] can be interpreted as a counting result similar to our Theorem 3.1 and [2, Theorems 1 and 3], though it is not phrased in the same language. Standard techniques would then yield a Jarník-type theorem which could then be compared with Theorem 2.1 and [2, Corollaries 3 and 5]. We leave the details to the interested reader.
Acknowledgements. The author was supported by the EPSRC Programme Grant EP/J018260/1. The author thanks Victor Beresnevich, Sanju Velani, and Evgeniy Zorin for helpful discussions. The author thanks the anonymous referee for helful comments.

A Jarník-type result
The Hausdorff-Cantelli lemma [3, Lemma 3.10] is a generalization of the Borel-Cantelli lemma and states that if g is a dimension function (i.e. a nondecreasing continuous function such that lim ρ→0 g(ρ) = 0) and (B( where H g denotes Hausdorff measure with respect to the gauge function g (cf. [7, §4.9]). As a special case, if ψ : N → [0, ∞) is a function such that the series converges, then for all θ ∈ R n , we have H g (S(ψ, θ)) = 0. As in the previous section, we will give a name to those manifolds that "inherit" this property from R n . Precisely, we will say that a manifold M ⊂ R n is of strong (resp. weak) Jarník type for convergence with respect to a dimension function g if for every function (resp. monotonic function) ψ such that the series (2.1) converges, we have Here m denotes the codimension of M. Intuitively, a manifold is of Jarník type for convergence if the "size of S(ψ, θ) ∩ M relative to M" is no bigger than the "size of S(ψ, θ) relative to R n ", as measured by the dimension function g. Note that a manifold is of strong (resp. weak) Khinchin type for convergence if and only if it is of strong (resp. weak) Jarník type for convergence with respect to the dimension function g(ρ) = ρ n .
rectangle, let f : K → R m be a function of class C 2 , and let the manifold M ⊂ R n be given by (1.2). Let g be a dimension function such that g is increasing, and suppose that for some k ∈ N, both of the following hold: (1) The series Then M is of strong Jarník type for convergence with respect to the dimension function g.
. On the other hand, So the convergence of (2.2) with k = 1 (resp. k = 2) is guaranteed by the appropriate numerical hypothesis of Case (1) (resp. Case (2)) of Theorem 1.3.

A counting result
The proof of Theorem 2.1 is based on a counting result that is interesting in its own right. Throughout this section, we fix d, m ∈ N, a closed rectangle K ⊂ R d , and a function f : K → R m of class C 2 . Now for each q ∈ N, κ > 0, and θ ∈ R n , we write θ = (λ, γ) ∈ R d × R m , we consider the set and we let A(q, κ, θ) = A K,f (q, κ, θ) = #R(q, κ, θ).

Convention.
The notation A B means that there exists a constant C ≥ 1 (the implied constant), depending only on universal variables such as d, m, K, and f (but not on q, κ, and θ), such that A ≤ CB. The notation A B means A B A. The notation A + B means that there exists Then for all q ∈ N, κ > 0, and θ ∈ R n , we have where φ(q) = (q −1 Log 2 (q)) k 2m+k . Here and hereafter we use the notation Log(q) def = max(1, log(q)).
Proof of Theorem 2.1 using Theorem 3.1. First consider the case where (2.3) holds for all α ∈ K. Then there exist a rectangle L ⊂ R d whose interior contains K and an extension of f to L such that (2.3) holds for all α ∈ L. Let C 1 = 1 + max α∈L |f (α)|, and fix ψ such that (2.1) converges. It is not hard to see that where the ball is taken with respect to the max norm. So by the Hausdorff-Cantelli lemma, if the series ∞ q=1 A L,f (q, C 1 ψ(q), θ) g ψ(q) q converges then H g (S(ψ, θ) ∩ M) = 0. And indeed, by Theorem 3.1, which completes the proof in this case.
For the general case, we proceed to re-use the argument given in [2, Step 2 on p. 17]: Let V be the set of points α ∈ K such that (2.3) holds. Since V is open, it can be written as the union of countably many rectangles, say

Proof of Theorem 3.1
The following lemma is a reformulation of the main technical result of [2]. We provide the proof for completeness.

Some manifolds of Khinchin type for convergence 183
Proof. In what follows we assume that H, r ≥ 1. Let e denote the 1-periodic exponential function e(x) = exp(2πix). We will need the following estimates, valid for all x ∈ R and H ∈ N: Here, the right-hand side of (4.2) is written using Iverson bracket notation, i.e.
[φ] is 1 if φ is true and 0 if φ is false. Now let A : R d → R m be a linear transformation and fix y ∈ R m . We have v∈Z d |v|≤r assuming that a+v+λ q ∈ K. So if δ is chosen small enough (depending on f ), then and thus since 2κ ≤ (2H) −1 ,
We are now ready to prove Theorem 3.1:   , t), there exists a neighborhood U = U (α 0 , t 0 ) ⊂ Ω of (α 0 , t 0 ) such that for all (α, t) ∈ U , we have Without loss of generality, we may assume that U is of the form U = Here (K i ) d 1 denote the factors of K, so that K = K 1 × · · · × K d . We can also assume that the sets V 1 , . . . , V d are intervals. Now since Ω = K×∂[−1, 1] m is compact, there exists a finite set F ⊂ Ω such that the collection {U (α 0 , t 0 ) : (α 0 , t 0 ) ∈ F } covers Ω. Now fix (α 0 , t 0 ) ∈ F , let the notation be as above, and let Fix t ∈ W and β ∈ V , and consider the map where (β, β) j def = β j for j ∈ J and (β, β) j def = β j for j ∈ J. By (4.6) and the convexity of C(α 0 , t 0 ), the map Φ β is invertible and its Jacobian determinant is bounded away from zero. So where λ S denotes Lebesgue measure on a set S, Φ * [µ] denotes the pushforward of a measure µ under a function Φ, and R > 0 is sufficiently large. By integrating with respect to β, we get where Φ(α) = (t · f (α)[e i ]) i∈I . Now fix q ∈ N, κ > 0, and θ ∈ R n , let δ > 0 and H, r ∈ N be as in Lemma 4.1, and assume that H, r ≥ 1. Fix h ∈ Z m such that 0 < |h| ≤ H.
Let η = |h| ≥ 1 and t = η −1 h, and fix (α 0 , t 0 ) ∈ F such that t ∈ W (α 0 , t 0 ). Letting the notation be as above, we have Taking the sum over all (α 0 , t 0 ) ∈ F such that t ∈ W (α 0 , t 0 ) gives The inequality κ ≥ φ(q) allows us to compare the two terms on the righthand side of (4.8): which shows that the right-hand term of (4.8) is smaller than the left-hand term. Thus A(q, κ, θ) q d κ m , and we have completed the proof in the case φ(q) ≤ κ ≤ 1/4, q sufficiently large.
, assuming q is large enough so that φ(q) ≤ 1/4. Thus, (3.1) holds in these cases as well. Finally, if q is bounded, then the right hand side of (3.1) is bounded from below while the right hand side is bounded from above, so (3.1) holds in this case as well.

Typicality of the condition (1.7)
The reader may notice that we did not use the hypothesis m < d+1 2 in the proof of Theorem 1.3, Case (2) and in particular U C 3 = ∅. Thus F C 3 is a proper algebraic subset of Ω. Since such a set has dimension strictly less than that of the ambient space, it is nowhere dense and thus U C 3 (and similarly U ) is dense. This completes the proof of (iv ).
Let W ⊂ R d be a nonempty open set such that W ∩ −W = ∅. Suppose that U 3 = ∅, and fix (Q 1 , . . . , Q ) ∈ U 3 . Then (Q 1 , . . . , Q ) : W → R is an injective continuous map. Since such a map cannot be dimensiondecreasing, we have ≥ d. This completes the proof of (v ).
Let B be a smooth point of S, and let T B S denote the tangent space of S at B. Suppose that < 2d − 1. Then there exists a subspace V 0 ∈ G m (Sym 2 R d ) intersecting S transversely at B. Here by "transversely" we mean that V 0 + T B S = Sym 2 R d ; we allow V 0 ∩ T B S to be nontrivial, and in fact necessarily dim(V 0 ∩T B S) ≥ 1 since B ∈ V 0 ∩T B S. If V ∈ G m (Sym 2 R d ) is sufficiently close to V 0 , then V ∩ S = {0}, so V / ∈ U 2 . So there is a neighborhood of V 0 disjoint from U 2 , proving (vi ).