Counting rational points close to p-adic integers and applications in Diophantine approximation
aa r X i v : . [ m a t h . N T ] F e b COUNTING RATIONAL POINTS CLOSE TO p -ADIC INTEGERS ANDAPPLICATIONS IN DIOPHANTINE APPROXIMATION BENJAMIN WARD
Abstract.
We find upper and lower bounds on the number of rational points that are ψ -approximations of some p -adic integer. Lattice point counting techniques are used to find theupper bound result, and a Pigeon-hole principle style argument is used to find the lower boundresult. We use these results to find the Hausdorff dimension for the set of p -adic weighted simulta-neously approximable points intersected with p -adic coordinate hyperplanes. For the lower boundresult we show that the set of rational points that τ -approximate a p -adic integer form a set ofresonant points that can be used to construct a local ubiquitous system of rectangles. Introduction
The study of rational points on algebraic varieties, usually called Diophantine geometry, has a widevariety of applications in many areas of mathematics. A variation of this is the study of rationalpoints that lie close to such algebraic varieties. In the setting of R n there has been many resultsof this type, including counts on the number of rational points close to curves [7, 37, 34, 35, 24]and manifolds [4, 13, 22, 23]. In the p -adic setting less is known. In [2, 3] a bound on the numberof rational points that lie on the curve C f = { ( x, x , . . . , x n ) : x ∈ Z p } were found, but as yet noother results are available. In this paper we provide an upper and lower bound on the numberof rational points within a small neighbourhood of a p -adic integer. Such result allows us to findbounds on the number of rational points close to p -adic coordinate hyperplanes.Fix a prime number p ∈ N and let | . | p denote the p -adic norm. Define the set of p -adic numbers Q p as the completion of Q with respect to the p -adic norm. Denote by Z p := { x ∈ Q p : | x | p ≤ } the ring of p -adic integers. Let x ∈ Z p , N ∈ N , and ψ : N → R + , with ψ ( q ) → q → ∞ . Weprovide bounds on the cardinality of the set Q ( x, ψ, N ) := ( ( q, q ) ∈ N × Z : 0 < q ≤ N, | q | ≤ N, (cid:12)(cid:12)(cid:12)(cid:12) x − q q (cid:12)(cid:12)(cid:12)(cid:12) p < ψ ( N ) ) . If the approximation function ψ is of the form ψ ( q ) = q − τ we will use the notation Q ( x, τ, N ).Note that to get a result for general x ∈ Z p we must apply some conditions on x ∈ Z p . Forexample, if x ∈ Q then for sufficiently large N ∈ N we have that Q ( x, ψ, N ) ≍ N , where by Date : January, 2020. ≍ b we mean there exists constants c , c ∈ R > such that c b ≤ a ≤ c b . Conversely, if x isbadly approximable and ψ ( q ) < q − − ǫ for some ǫ > Q ( x, ψ, N ) ≪
1. In order to obtaingood bounds on the cardinality of Q ( x, ψ, N ) we use the Diophantine exponent, τ ( x ), where τ ( x ) := sup { τ > | qx − q | p < Q − τ , for i.m. Q ∈ N , with 0 < q, | q | ≤ Q, } . By a Theorem of Mahler [36] we have that for all x ∈ Z p , τ ( x ) ≥
2. Further, by a result of Jarnik[27] we have that τ ( x ) ≤ x ∈ Z p , with respect to the Haar measure µ p on Q p ,normalised by µ p ( Z p ) = 1.We have the following results on the cardinality of Q ( x, ψ, N ). Lemma 1.1.
Let x ∈ Z p with Diophantine exponent τ ( x ) and let ψ ( q ) = q − τ for some τ ∈ R + with max { , τ ( x ) − } < τ < τ ( x ) . Then for any ǫ > there exists sufficiently large Q ∈ N suchthat Q ( x, τ, Q ) ≤ N τ ( x ) − τ + ǫ . Note by our previous remark on the Diophantine exponent that for almost all x ∈ Z p if ψ ( q ) = q − τ with 1 < τ <
2, then Q ( x, ψ, Q ) ≤ Q − τ + ǫ . While Lemma 1.1 gives us an upper bound for all x ∈ Z p , provided the approximation function ψ is ’close’ to the function related to the Diophantine exponent, the bound given has an extra Q ǫ term, which we believe is unnecessary. The following theorem offers an improvement in thisrespect. Theorem 1.2.
Let x ∈ Z p and suppose that τ ( x ) = 2 . Let ψ : N → R + be an approximationfunction with q − < ψ ( q ) < q − . Then for sufficiently large M ∈ N , Q ( x, ψ, M ) ≤ M ψ ( M ) . Again, as with Lemma 1.1, we can deduce that the above upper bound is true for almost all x ∈ Z p . This type of result has already been proven in the real case (see Lemma 6.1 of [8]). Lastly,we have the following lemma which provides a complimentary lower bound to the previous tworesults. Lemma 1.3.
Let x ∈ Z p and < τ < . Then for sufficiently large Q ∈ N we have that Q ( x, τ, Q ) ≥ p Q − τ − . s with Theorem 1.2, the Euclidean version of this result has previously been proven, (see Lemma3 of [33]). Further, as τ < Q large enough such that Q ( x, τ, Q ) ≥ p Q − τ . Thus combining this with Theorem 1.2 we have the expected result that Q ( x, τ, N ) ≍ N − τ .The proofs of Lemma 1.1 and Lemma 1.3 use elementary techniques. The proof of Theorem 1.2 ismore substantial and uses p -adic approximation lattices and lattice counting techniques. Prior tothe proofs of these results we give an example of their applications in Diophantine approximation.2. p -adic Diophantine approximation As an application of the main results in the previous section we consider the set of p -adic simultane-ously approximable points over coordinate hyperplanes. Define the set of weighted simultaneouslyapproximable points as follows. For an n -tuple of approximation functions Ψ = ( ψ , . . . , ψ n ), with ψ i : N → R + for 1 ≤ i ≤ n , and q ∈ N let A q (Ψ) = [ | q i |≤ q ≤ i ≤ n (cid:8) x ∈ Z np : | qx i − q i | p < ψ i ( q ) (cid:9) , where x = ( x , . . . , x n ). Define the set of weighted Ψ-approximable p -adic points as W n (Ψ) := lim sup q →∞ A q (Ψ) . If the approximation functions have the form ψ ( q ) = q − τ i for some exponents of approximation τ = ( τ , . . . , τ n ) ∈ R n + we will use the notation W n ( τ ) = W n (Ψ). By considering the Dirichletstyle theorem for the set W n ( τ ) we have that W n ( τ ) = Z np provided that P ni =1 τ i ≤ n + 1. Let µ p,n denote the Haar measure of Q np , normalized by µ p,n ( Z np ) = 1. Jarnik [27] showed that for ψ monotonic decreasing µ p,n ( W n ( ψ )) has zero measure when(1) ∞ X q =1 ψ ( q ) n converges, and has full measure when (1) diverges. There are also developments of Jarnik’sTheorem to the weighted case [11] and linear forms [29].For sets of zero Haar measure we use Hausdorff measure and Hausdorff dimension to provide moreaccurate notions of size. We briefly recap the definition and notation of Hausdorff measure anddimension. For a metric space ( X, d ), a set U ⊂ X , and ρ >
0, define a ρ -cover of U as a sequenceof balls { B i } such that U ⊂ S i B i and for all balls r ( B i ) ≤ ρ , where r ( . ) denotes the radius of he ball. Define a dimension function f : R + → R + as an increasing continuous function with f ( r ) → r →
0. Define the f -Hausdorff measure as H f ( U ) = lim ρ → + inf (X i f ( r ( B i )) : { B i } is a ρ -cover of U ) , where the infimum is taken over all ρ -covers of U . When the dimension function f ( x ) = x s wewill use the notation H f = H s . Define the Hausdorff dimension asdim U = inf { s ≥ H s ( U ) = 0 } . In [11] it was proven that, for P ni =1 τ i > n + 1,dim W n ( τ ) = min ≤ i ≤ n ( n + 1 + P nj = i ( τ i − τ j ) τ i ) . Correspondingly to results over Z p it would be desirable to obtain measure results for simultaneous p -adic approximable points over manifolds. In [28] Kleinbock and Tomanov proved the extremalityof p -adic manifolds provided some non-degeneracy conditions are satisfied. Generally a manifold M ⊂ Z np is said to be extremal if almost all points, with respect to the induced Haar measureof the manifold, we have that τ ( x ) = n + 1 /n (see [28] for more details). There are a varietyof results for p -adic dual approximation, see for example [5, 9, 15, 18, 30], however results in thesimultaneous case are lacking. Recently Oliveira [31] produced a Khintchine-style Theorem forsimultaneous p -adic approximation with denominators coming from p -adic balls. This result has asimilar style to our result, with the difference being that our denominators come from a ball withradius tending to zero, rather than a fixed constant. Other than this result there are relativelyfew Khintchine-style results.For the Hausdorff dimension there are recent results on simultaneously approximable points over C f = { ( x, x , . . . , x n : x ∈ Z p } for sufficiently large Diophantine exponents [3, 16]. In the previouschapter a lower bound for the Hausdorff dimension was found for general n -dimensional normalcurves. A key reason the upper bound could not be obtained was a lack in results on the behaviourof rational points close to p -adic curves. The results of the previous section give us a goodunderstanding of the behaviour of rational points close to coordinate hyperplanes. The results ofthis chapter are closely related to a variety of results in the real case on Diophantine approximationover coordinate hyperplanes, see [10, 32, 33].For a p -adic integer α ∈ Z p and 1 ≤ m ≤ n define the coordinate hyperplaneΠ α,m := { ( x , . . . , x m − , α, x m +1 , . . . , x n ) : ( x , . . . , x m − , x m +1 , . . . , x n ) ∈ Z n − p } ⊂ Z np . or the set W n ( τ ) ∩ Π α,m we have the trivial result thatdim W n ( τ ) ∩ Π α,m ≤ dim Π α,m = n − , with equality when P ni =1 τ i ≤ n + 1. In this paper we prove the following result on the Hausdorffdimension of W n (Ψ) ∩ Π α . Theorem 2.1.
Let Π α,m be a coordinate hyperplane of Z np , let α ∈ Z p \ Q , and ≤ m ≤ n . Let τ = ( τ , . . . , τ n ) ∈ R n + be a weight vector with the properties that < τ m < , and n X i =1 τ i > n + 1 . Then for almost all α ∈ Z p \ Q (with respect to the Haar measure), dim W n ( τ ) ∩ Π α,m = min ≤ i ≤ ni = m n + 1 − τ m + P τ j ≤ τ i j = m ( τ i − τ j ) τ i . Remark 2.2.
The constraints on τ m ensure that we can apply Theorem 1.2. Note that we canuse the same style of proof used to prove the upper bound of 2.1, in combination with Lemma 1.1,to prove that for max { , τ ( α ) − } < τ m < τ ( α ) we have thatdim W n ( τ ) ∩ Π α,m ≤ min ≤ i ≤ ni = m n + τ ( α ) − − τ m + P τ j ≤ τ i j = m ( τ i − τ j ) ,τ i . Proving the corresponding lower bound of this result is currently beyond our reach.
Remark 2.3.
The lower bound over all τ i ensures that we do not include the trivial case when P ni =1 τ i ≤ n + 1, in which case W n ( τ ) has full dimension and so dim W n ( τ ) ∩ Π α,m = n −
1. Inthe special case where the approximation functions are the same i.e. ( τ = ( τ, . . . , τ )), then wehave that, for 1 + n ≤ τ <
2, dim W n ( τ ) ∩ Π α = n + 1 τ − . Note that this gives us the expected dimension of the set of approximable points W n ( τ ) less thecodimension of the hyperplane Π α,m . or general approximation functions Ψ = ( ψ , . . . , ψ n ), providing the limit(2) ψ ∗ i = lim q →∞ − log( ψ ( q ))log q exists and is positive finite for each 1 ≤ i ≤ n , define Ψ ∗ = ( ψ ∗ , . . . , ψ ∗ n ). Then we have thefollowing corollary. Corollary 2.4.
Let
Ψ = ( ψ , . . . , ψ n ) with ψ i : N → R + for each ≤ i ≤ n be an approximationvector with each ψ i having positive finite limit (2) . If Ψ ∗ satisfy the same conditions as Theorem2.1, then for almost all α ∈ Z p \ Q , dim W n (Ψ) ∩ Π α = min ≤ i ≤ ni = m ( n + 1 − ψ ∗ m + P nj = i, j = m ( ψ ∗ i − ψ ∗ j ) ψ ∗ i ) . The corollary easily follows from the observation that W n (Ψ ∗ + ǫ ) ⊆ W n (Ψ) ⊆ W n (Ψ ∗ − ǫ )for any ǫ >
0. As such limit (2) exist for each 1 ≤ i ≤ n we can let ǫ → Auxiliary results
We provide a brief set of known results that we will use in the proof of Theorem 2.1. The firstresult we state can be considered as the p -adic version of Minkowski’s theorem for systems of linearforms. The proof is a straightforward application of the pigeon-hole principle and can be found in[11]. Lemma 3.1.
Let L i ( x ) , with i = 1 , . . . , n , be linear forms with p -adic coefficients of x =( x , x , . . . , x n ) . Let P ni =1 τ i = n + 1 for τ i ∈ R + , and H ≥ . Then there exists a non-zerorational integer vector x = ( x , x , . . . , x n ) with max ≤ i ≤ n | x i | ≤ H satisfying the system of inequalities | L i ( x ) | p < pH − τ i for i = 1 , . . . , n. he following lemma generally states that the measure of a lim sup set of balls remains unalteredwhen the radius is multiplied by some constant. The Euclidean version of this result is well knownand appears in a variety of texts, see [6]. The following version for ultrametric spaces was provenin [11]. Lemma 3.2.
Let ( X, d ) be a separable ultrametric space and µ be a Borel regular measure on X .Let ( B i ) i ∈ N be a sequence of balls in X with radii r i → as i → ∞ . Let ( U i ) i ∈ N be a sequence of µ -measurable sets such that U i ⊂ B i for all i . Assume that for some c > | U i | ≥ c | B i | for all i . Then the limsup sets U = lim sup i →∞ U i := ∞ T j =1 S i ≥ j U i and B = lim sup i →∞ B i := ∞ T j =1 S i ≥ j B i have the same µ -measure. In particular, if we chose the approximation function ψ i ( q ) = pq /n for each 1 ≤ i ≤ n then byLemma 3.1 we know W n (Ψ) = Z p , and so by shrinking the lim sup set of balls by constant 1 /p Lemma 3.2 gives us that µ n ( W n (Ψ /p )) = 1.Another key result in our proof of Theorem 2.1 is the following Mass Transference Principle typetheorem. In order to state this theorem we need the notion of local ubiquity for rectangles, avariation of the notion of ubiquity introduced by Beresnevich, Dickinson, and Velani [6]. Fix aninteger n ≥
1, and for each 1 ≤ i ≤ n let ( X i , | . | i , m i ) be a bounded locally compact metric spacewith m i a δ i -Ahlfors probability measure. Consider the product space ( X, | . | , m ), where X = n Y i =1 X i , m = n Y i =1 m i , | . | = max ≤ i ≤ n | . | i . For any x ∈ X and r ∈ R + define the open ball B ( x, r ) = (cid:26) y ∈ X : max ≤ i ≤ n | x i − y i | i < r (cid:27) = n Y i =1 B i ( x i , r ) , where B i are the usual balls associated with the i th metric space. Let J be a infinite countableindex set, and β : J → R + a positive function. Let l n , u n be two sequences in R + such that u n ≥ l n with l n → ∞ as n → ∞ . Define J n = { α ∈ J : l n ≤ β α ≤ u n } . et ρ : R + → R + be a non-increasing function with ρ ( β α ) → β α → ∞ . For each 1 ≤ i ≤ n ,let { R α,i } α ∈ J be a sequence of subsets in X i . As with the standard setting of ubiquitous systemsdefine the resonant sets ( R α = n Y i =1 R α,i ) α ∈ J . For a = ( a , . . . , a n ) ∈ R n + denote the set of hyperrectangles∆( R α , ρ ( r ) a ) = n Y i =1 ∆( R α,i , ρ ( r ) a i ) , where for some set A and b ∈ R + ∆( A, b ) = [ a ∈ A B ( a, b ) . Definition 3.3 (local ubiquitous system of rectangles) . Call ( { R α } α ∈ J , β ) a local ubiquitous systemof rectangles with respect to ( ρ, a ) if there exists a constant c > such that for any ball B ⊂ X , lim sup n →∞ m B ∩ [ α ∈ J n ∆( R α , ρ ( U n ) a ) ! ≥ cm ( B ) . The second property needed to state the theorem is the following local scaling property, firstintroduced in [1]. While we will not need it in our use of the Theorem 3.5 we state it and includeit in the final theorem for completeness.
Definition 3.4 ( k -scaling property) . Let ≤ k < and ≤ i ≤ n . Then { R α,i } α ∈ J has k -scalingproperty if for any α ∈ J , any ball B ( x i , r ) ⊂ X i with centre x i ∈ R α,i , and < ǫ < r then c r δ i k ǫ δ i (1 − k ) ≤ m i ( B ( x i , r ) ∩ ∆( R α,i , ǫ )) ≤ c r δ i k ǫ δ i (1 − k ) , for some constants c , c > . In our use the resonant sets will be sets of points, so k = 0. For t = ( t , . . . , t n ) ∈ R n + define W ( t ) = lim sup α ∈ J ∆ (cid:0) R α , ρ ( β α ) a + t (cid:1) . Given the above notations and definitions we can state the Mass Transference Principle fromrectangles to rectangles (MTPRR) of [38].
Theorem 3.5 (Mass Transference Principle from rectangles to rectangles) . Under the settingsabove assume that ( { R α } α ∈ J , β ) satisfies the local ubiquity for systems of rectangles condition with espect to ( ρ, a ) , and the k -scaling property. Then dim W ( t ) ≥ min A i ∈ A ( X j ∈ K δ j + X j ∈ K δ j + k X j ∈ K δ j + (1 − k ) P j ∈ K a j δ j − P j ∈ K t j δ j A i ) , where A = { a i , a i + t i , ≤ i ≤ n } and K , K , K are a partition of { , . . . , n } defined as K = { j : a j ≥ A i } , K = { j : a j + t j ≤ A i }\ K , K = { , . . . n }\ ( K ∪ K ) . Hence, provided we can find a lim sup set of hyperrectangles that satisfy the local ubiquity prop-erty for rectangles then we have a lower bound for the corresponding lim sup set of shrunkenhyperrectangles. 4.
Proof of Theorem 2.1
We split the proof into the upper and lower bound, and solve each case separately. In both caseswe will use the projection π : Z np → Z n − p , defined by( x , . . . , x n ) ( x , . . . , x m − , x m +1 , . . . , x n ) . By a well known theorem of Hausdorff theory (see Proposition 3.3 of [20]) as π is a bi-Lipschitzmapping over W n ( τ ) ∩ Π α,m , we have thatdim W n ( τ ) ∩ Π α,m = dim π ( W n ( τ ) ∩ Π α,m )) . Consider the set of integers Q ( α, τ m ) := { q ∈ N : | qα − q m | p < q − τ m , for some | q m | ≤ q } , and the union of sets A ∗ q ( τ ) = [ | q i |≤ q ≤ i ≤ ni = m (cid:8) x ∈ Z n − p : | qx i − q i | p < q − τ i (cid:9) . Then, π ( W n ( τ ) ∩ Π α,m ) = lim sup q ∈Q ( α,τ m ) A ∗ q ( τ ) , hence we only need to find the upper and lower bounds for dim lim sup q ∈Q ( α,τ m ) A ∗ q ( τ ). .1. Upper bound.
For the upper bound we take the standard cover of hyperrectangles usedin the construction of lim sup q ∈Q ( α,τ m ) A ∗ q ( τ ). By a standard geometrical argument we note that eachhyperrectangle, centred at some q q ∈ Q in the construction of A ∗ q ( τ ), can be covered by a finitecollection of balls B q ( τ i ) of radius q − τ i , ( i = m ). Without loss of generality we can assume that τ ≥ · · · ≥ τ m − ≥ τ m +1 ≥ · · · ≥ τ n , since if not then we could take some bi-lipschitz mapping to reorder the coordinate axis such thatthis was the case. Hence for each j ≤ i , q − τ j q − τ i ≤ . Hence in the product below we only consider the j ≥ i . By the above argument we have that thecardinality of B q ( τ i ) is B q ( τ i ) ≪ n Y j = ii = m q − τ j q − τ i = q P nj = i, j = m ( τ i − τ j ) . As each τ i -approximation function is decreasing as q increases, for each interval 2 k ≤ q < k +1 take q = 2 k . Using that Q ( α, τ m ) ⊆ S k ∈ N Q ( α, τ m , k ), then H s lim sup q ∈Q ( α,τ m ) A ∗ q ( τ ) ! ≤ ∞ X k =1 X q ∈Q ( α,τ m , k ) ( q ) n − B q ( τ i ) . ( q − τ i ) s , ( a ) ≪ ∞ X k =1 k (2 − τ m ) (2 k ) n − (2 k ) P nj = i, j = m ( τ i − τ j ) (2 k ) − τ i s , ≪ ∞ X k =1 k ( n +1 − τ m + P nj = i, j = m ( τ i − τ j ) − τ i s ) , where ( a ) follows from Theorem 1.2. The above sum converges when s ≥ n + 1 − τ m + P nj = i, j = m ( τ i − τ j ) τ i + ǫ, for any ǫ >
0. This is true for each 1 ≤ i ≤ n , i = m , and as ǫ is arbitrary, we have that s ≥ min ≤ i ≤ ni = m ( n + 1 − τ m + P nj = i, j = m ( τ i − τ j ) τ i ) , completing the upper bound result. Note that the result of Remark 2.2 can similarly be obtainedby replacing Theorem 1.2 used at ( a ) by Lemma 1.1. .2. Lower bound.
In order to use Theorem 3.5 to prove the lower bound of Theorem 2.1 weneed to construct a ubiquitous system of rectangles. In following with the ubiquity setup forTheorem 3.5 let J = Q ( α, τ m ) , R q,i = n q i q ∈ Q : | q i | ≤ q o , R q = Q ni =1 i = m R q,i ,β ( q ) = q, ρ ( q ) = q − , l k = M k , u k = M k +1 , where M ∈ N is a fixed integer to be determined later. Then we have that J k = { q ∈ N : M k ≤ q < M k +1 : ∃| q m | ≤ q s.t. | qα − q m | p < M − ( k +1) τ m } . Hence for some vector a = ( a , . . . , a n ) ∈ R n + ,∆( R α , ρ ( r ) a ) = n Y i =1 [ | q i |≤ q B (cid:18) q i q , r − a i (cid:19) . We prove the following.
Proposition 4.1.
Let R q , ρ , and J k be as above, and let ˜ v = ( v , . . . , v m − , v m +1 , . . . , v n ) ∈ R n + with each v i > and n X i =1 i = m v i = n + 1 − τ m , where < τ m < . Then for any ball B = B ( x, r ) ⊂ Z n − p , with centre x ∈ Z n − p and radius < r < r for some r ∈ R + , there exists a constant c > such that µ n − B ∩ [ q ∈ J k ∆( R q , ρ ( u k ) ˜ v ) ! ≥ cµ n − ( B ) , provided M ≥ n − . The proof of this result follows the same style of many similar results. For example see Theorem1.3 of [12] for the one dimensional real case, or Proposition 5.1 of [11] for the n -dimensional p -adiccase. Proof.
For any y = ( y , . . . , y m − , y m +1 , . . . , y n ) ∈ ( Z p \ Q ) n − , consider the system of inequalities(3) | qα − q m | p < ( M k +1 ) − τ m , | qy i − q i | p < ( M k +1 ) − v i , ≤ i ≤ n, i = m, max ≤ i ≤ n {| q | , | q i |} ≤ M k +1 . By the condition on ˜ v we have, by Lemma 3.1, that there exists a non-zero integer solution( q , q ) ∈ Z n × Z to (3). Assume without loss of generality that q ≥ = 0. As v i > q i = 0, whichcontradicts the fact that ( q , q ) is a non-zero solution, thus q >
0. Further, if ( q , q ) solves (3) then q ∈ Q ( α, τ m , M k +1 ).Thus, we have that µ n − B ∩ [ q ∈Q ( α,τ m ,M k +1 ) ∆( R q , ρ ( M k +1 ) ˜ v )) = µ n − ( B ) . Note that µ n − B ∩ [ q ∈Q ( α,τ m ,M k +1 ) ∆( R q , ρ ( M k +1 ) ˜ v )) ≤ µ n − B ∩ [ q ∈Q ( α,τ m ,M k ) ∆( R q , ρ ( M k +1 ) ˜ v )) + µ n − B ∩ [ q ∈ J k ∆( R q , ρ ( M k +1 ) ˜ v )) ! , and so µ n − B ∩ [ q ∈ J k ∆( R q , ρ ( M k +1 ) ˜ v )) ! ≥ µ n − ( B ) − µ n − B ∩ [ q ∈Q ( α,τ m ,M k ) ∆( R q , ρ ( M k +1 ) ˜ v )) . At this point we only want the q q ∈ R q such that B ∩ B (cid:18) q q , ρ ( M k +1 ) ˜ v (cid:19) = ∅ . For ball B = B ( x, r ) with x ∈ Z n − p and r ∈ { p j : j ∈ Z } , this is equivalent to the set of solutionsto(4) (cid:12)(cid:12)(cid:12)(cid:12) x − q i q (cid:12)(cid:12)(cid:12)(cid:12) p < r, ≤ i ≤ n, i = m. For q fixed and each | q i | ≤ q by residue classes we have that there are at most(2 qr + 1) n − suitable values of q . We can choose suitably large k ∈ N such that M k r >
1, and so for each q, | q i | ≤ M k there are at most(5) (3 M k r ) n − ossible values of q solving (4). Hence µ n − B ∩ [ q ∈Q ( α,τ m ,M k ) ∆( R q , ρ ( M k +1 ) ˜ v )) ≤ X q ∈Q ( α,τ m ,M k ) X q solving (4) µ n − (cid:18) B ∩ ∆ (cid:18) q q , ρ ( M k +1 ) ˜ v ) (cid:19)(cid:19) , ( a ) ≤ X q ∈Q ( α,τ m ,M k ) (3 M k r ) n − ( M k +1 ) − ( n − P ni =1 i = m v i , ( b ) ≤ M k (2 − τ m ) n − M k ( n − M − ( k +1)( n +1 − τ m ) µ n − ( B ) , ≤ . n M − n − τ m µ n − ( B ) , where ( a ) follows by (5) and ( b ) follows by Theorem 1.2 and our condition on ˜ v . As M ≥ n − c = (cid:18) − . n M n +1 − τ m (cid:19) > . Thus, µ n − B ∩ [ q ∈ J k ∆( R q , ρ ( M k +1 ) ˜ v )) ! ≥ c.µ n − ( B ) . (cid:3) Given Proposition 4.1 we have that ( R q , β ) is a local ubiquitous system of rectangles with respectto ( ρ, ˜ v ), provided P ni =1 i = m v i = n + 1 − τ m . Without loss of generality assume that m = n , and so˜ v = ( v , . . . , v n − ) ∈ R n − . Given τ = ( τ , . . . , τ n ) ∈ R n + assume without loss of generality that τ > τ > · · · > τ n − and define each v n − i recursively by v n − i = min ( τ n − i − ε, n + 1 − τ n − P n − j = n − i v i n − i ) . By the condition on τ of Theorem 2.1, there exists a k ∈ { , . . . , n − } such that v j = n + 1 − τ n − P n − j = n − k v i n − − k , for all 1 ≤ j ≤ k . Clearly each v i < τ i for 1 ≤ i ≤ n −
1, and so the associated vector t = ( t , . . . t n ) ∈ R n − is defined by t i = τ i − v i , ≤ i ≤ n − . Consider the set A = { v , . . . , v n − , τ , . . . , τ n − } . For each A i ∈ A observe the following: ) A i ∈ { v , . . . , v n − } : Then we have the sets K = { , . . . , max { i, n − k }} , K = { max { i + 1 , n − k + 1 } , . . . , n − } , K = ∅ . By Theorem 3.5 we have thatdim W n ( τ ) ≥ min A i ( max { i, n − k } v i + ( n − max { i + 1 , n − k + 1 } ) v i − P n − j =max { i +1 ,n − k +1 } t j v i ) , = min A i ( ( n − v i − P n − j =max { i +1 ,n − k +1 } t j v i ) . Since t j = ε for n − k < j ≤ n − W n ( τ ) = n − A i ∈ { τ , . . . , τ n − } : Since τ i = v i + ε for n − k + 1 ≤ i ≤ n − τ i for 1 ≤ i ≤ n − k . For such τ i we have the sets K = ∅ , K = { i, . . . , n − } , K = { , . . . , i − } . Applying Theorem 3.5 we havedim W n ( τ ) ≥ min A i ( ( n − i ) τ i + P i − j =1 v j − P n − j = i t j τ i ) , = min A i ( n − i ) τ i + ( n − k ) (cid:18) n +1 − τ n − P n − j = n − k v j n − k (cid:19) − P n − kj =1 τ j τ i , = min A i ( n + 1 − τ n + P n − j = i ( τ i − τ j ) τ i ) . Combining i) and ii), and returning to the τ m approximation over the α coefficient, we have thatdim lim sup q ∈Q ( α,τ m ) A ∗ q ( τ ) ≥ min ≤ i ≤ ni = m ( n + 1 − τ m + P nj = i, j = m ( τ i − τ j ) τ i ) , completing the proof. 5. Proof of the counting results
Recall, we are aim to provide bounds on the set Q ( x, ψ, N ) := ( ( q, q ) ∈ N × Z : 0 < q ≤ N, | q | ≤ N, (cid:12)(cid:12)(cid:12)(cid:12) x − q q (cid:12)(cid:12)(cid:12)(cid:12) p < ψ ( N ) ) . e begin with the proof of Lemma 1.3. This style of proof is not new and follows a similar ideato Euclidean version (see Lemma 3 of [33]). Proof of Lemma 1.3:
Fix x ∈ Z p and take t ∈ N to be the integer such that p − t ≤ Q − τ < p − t +1 . Consider a set of open disjoint balls { B i } p t i =1 , each with some centre point k i ∈ Z and radius p − t .Choose the set of points k i such that Z p ⊆ S p t i =1 B i . Consider the ( Q + 1) set of points of theform qx − q ∈ Z p , for ( q, q ) ∈ [0 , Q ] . By the Pigeon-hole principle there exists at least one ball, say B j , such thatthere are at least ( Q + 1) p t > p Q − τ points. As τ < Q sufficiently large enough such that p − Q − τ >
2. Order thepoints ( q, q ), that correspond to the set of points qx − q contained in B j , in terms of the absolutevalue of the q component. Suppose that the pair ( m, m ) is the smallest. Then for all other pairs,say ( q, q ), with corresponding point contained in B j we have that | ( q − m ) x − ( q − m ) + k j − k j | p < p − t . Clearly ( q − m ) ∈ (0 , Q ], and ( q − m ) ∈ [ − Q, Q ]. Note that the above argument yields p − Q − τ − (cid:3) Lemma 1.1 is also a relatively simple proof. We are unable to find a proof that uses a similarargument, however we suspect such style of result has been used before.
Proof of Lemma 1.1
We use a proof by contradiction. Suppose that(6) Q ( x, τ, Q ) > Q τ ( x ) − τ + ǫ . We use the following notations. Let X ∈ N be an integer such that | x − X | p < p − M , for some suitably large M ∈ N . Define V + Q and V − Q to be the sets V + Q := { ( q, q ) ∈ N × Z : 0 < q ≤ Q, ≤ q ≤ Q, } ,V − Q := { ( q, q ) ∈ N × Z : 0 < q ≤ Q, − Q ≤ q ≤ , } . Let t ∈ N be the integer such that p − t ≤ Q − τ < p − t +1 , nd similarly k ∈ N be the integer such that p − k ≤ Q − ( τ ( x )+ ǫ ) < p − k +1 . Note that as τ ( x ) > τ , we have that k ≥ t , and so p k − t ∈ N . Further, observe that(7) p k − t < pQ τ ( x ) − τ + ǫ . Lastly, by the definition of τ ( x ), we have that there exists finitely many Q ∈ N such that | qx − q | p < Q − ( τ ( x )+ ǫ ) , for 0 < q, | q | ≤ Q . Hence our ’sufficiently large Q ’ is the value of Q such that for any pair0 < q, | q | ≤ Q ,(8) | qx − q | p ≥ Q − ( τ ( x )+ ǫ ) , for all ǫ >
0. Consider the set of points in Q ( Q, τ, x ). Note that ( q, q ) ∈ Q ( Q, τ, x ) if and only if( q, q ) ∈ V + Q ∪ V − Q , and(9) qX − q ≡ p t . Thus, for all ( q, q ) ∈ Q ( Q, τ, x ) we have that qX − q = λp t , for some λ ∈ Z . Split the set of points in Q ( x, τ, Q ) into two disjoint sets, the set of pairs in V + Q ,and the set of pairs in V − Q . As there are greater than 2 Q τ ( x ) − τ + ǫ pairs, at least one of the setshas greater than Q τ ( x ) − τ + ǫ pairs. Without loss of generality assume such set of points belong in V + Q . Considering the range of values of λp t there are p k − t possible values of λp t modulo p k . By(6) and (7) we have, by the Pigeon-hole principle, that there exists at least two pairs, say ( a, a )and ( b, b ), such that ( a − b ) X − ( a − b ) ≡ p k . This is equivalent to | ( a − b ) x − ( a − b ) | p ≤ p − k ≤ Q − ( τ ( x )+ ǫ ) , with ( a − b, a − b ) ∈ V + Q ∪ V − Q , as 0 < a − b ≤ Q by our choice of ordering of a, b , and | a − b | ≤ Q by the fact that the pairs ( a, a ) , ( b, b ) ∈ V + Q . However, such result contradicts (8) which followsfrom the definition of τ ( x ), thus (6) must be false. (cid:3) .1. p -adic approximation lattices. Prior to the proof of Theorem 1.2 we recall some basicdefinitions and results of Lattice theory that will be needed. Define a lattice Λ as a discreteadditive subgroup of R n . If Λ ⊆ Z n the Λ is an integer lattice. A set of linearly independentvectors b , . . . , b n that generate Λ is called a basis of Λ. Let B be a n × n matrix with columns b i ,then call B a basis matrix. Define the fundamental region as F ( B ) := ( n X i =1 a i b i : a i ∈ R , ≤ a i < ) . A standard result of Lattice theory states that if B is a basis matrix for Λ then F ( B ) contains nolattice points other than the origin (see Chapter 3, Lemma 6 of [17]).The volume of the fundamental region can be found by taking the determinant of the basis matrix,that is vol ( F ( B )) = | det B | . A basis matrix is not unique for each Λ, however for any lattice Λthe volume of the fundamental region is the same regardless of choice of basis matrix. For thisreason the notation vol ( F ( B )) = | det Λ | is used. If U ∈ Z n × n is a unimodular matrix and B is abasis matrix for Λ then B = B U is also a basis matrix for Λ.One property of lattices that are incredibly useful are the successive minima of a lattice. Let B n = B (0 ,
1) denote the n -dimensional unit sphere. For c ∈ R + we use the notation cB n = B (0 , c ).Define the successive minima of a lattice Λ ⊂ R n of rank n as the set of values λ i (Λ) := min { λ > ∩ λB ) ≥ i } , for i = 1 , . . . , n . By Minkowski’s inequalities on the successive minima (see e.g. [21]) we have that(10) 2 n n ! det Λ ≤ vol ( B n ) n Y i =1 λ i (Λ) ≤ n det Λ . For a count on the number of lattice points within a convex body we have the follow theorem dueto Blichfeldt [14].
Theorem 5.1.
Let Λ ⊂ R n be a lattice of rank n and let V ⊂ R n be a convex body such that rank (Λ ∩ V ) = n . Then ∩ V ) ≤ n ! vol ( V )det Λ + n. The constant for such estimate can be excessively large, however in our use of the Theorem thesize of such constant is irrelevant.For the proof of Theorem 1.2 we use p -adic approximation lattices. First discovered by de Weger[19] who used them to prove a variety of results in classical p -adic Diophantine approximation, ncluding the p -adic analogue of Hurwitz Theorem. Recently n -dimensional forms of p -adic approx-imation lattices have been used to provided lattice based cryptosystems [25, 26]. In these papersboth dual and simultaneous approximation lattices were discussed. In particular Dirichlet-styleexponents were proven for simultaneous and dual approximation.For a n -tuple of approximation functions Ψ = ( ψ , . . . , ψ n ), a large integer M ∈ N , and a fixed x = ( x , . . . , x n ) ∈ Z np define the Ψ-approximation lattice Λ M byΛ M = { ( a , . . . , a n ) ∈ Z n +1 : | a x i − a i | p ≤ ψ i ( M ) , ≤ i ≤ n } . For any x ∈ Z np we may write each x j as the p -adic expansion x j = ∞ X i =0 x j,i p i , x j,i ∈ { , , . . . , p − } . Let X j,M ∈ Z be the integer X j,M = t j X i =0 x j,i p i , where each t j ∈ N is the unique value associated with M satisfying p − t j ≤ ψ j ( M ) < p − t j +1 . Lastly, for each 1 ≤ j ≤ n let ψ ∗ j,M = p t j . Then the set of vectors(11) B = X ,M ... X n,M , ψ ∗ ,M ...0 , . . . , ψ ∗ n,M , form a basis for Λ M . The set B can be proven to be a basis by considering the fundamental region F ( B ) and showing the only lattice point contained is . Given such basis we can deduce that | det Λ M | = n Y i =1 ψ ∗ i,M ≍ n Y i =1 ψ i ( M ) ! − . In the simultaneous case, Ψ = ( ψ, . . . , ψ ), it was proven in [26] that λ (Λ M ) ≪ ψ ( M ) − nn +1 . In order to prove Theorem 1.2 we find a lower bound on λ (Λ M ) by only considering x ∈ Z p satisfying certain Diophantine exponent properties. Proof of Theorem 1.2 : or approximation function ψ : N → R + with ψ ( q ) → q → ∞ , fixed x ∈ Z p and suitablylarge M ∈ N let ψ ∗ M and X M be defined as above. As x = 0 and x Q we can choose suitablylarge M such that | X M | >
2. For example, suppose that X M = − M . Then as M → ∞ ,by our definition of ψ ∗ M , t → ∞ . Thus X M = p − t X i =1 ( p − p i → − ∈ Z , as t → ∞ . As ψ ( q ) < q − we can choose suitably large M such that ψ ∗ M > M . Further, M is chosen suitablylarge enough such that for any ǫ > | q x − q | p ≥ ( M / − ǫ ) ǫ for all 0 ≤ q , | q | ≤ M / − ǫ . Note that such M is possible since τ ( x ) ≤ X M and ψ ∗ M we have that B = ( X M ! , ψ ∗ M !) form a basis for the lattice Λ M whereΛ M = { ( a , a ) ∈ Z : | a x − a | p < ψ ( M ) } . Further, we have that det Λ M = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X M ψ ∗ M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ψ ∗ M . Let V be the convex body V = [0 , M ] × [ − M, M ]. Consider the following two cases:i) rank (Λ M ∩ V ) = 2: By Theorem 5.1 we have that M ∩ V ≤ vol ( V )det Λ M + 2 , ≤ M . ( ψ ∗ M ) − + 2 , ≤ M ψ ( M ) , where the last inequality holds due to the condition that ψ ( q ) < q − . This proves Theorem 1.2for the rank 2 case.ii) rank (Λ M ∩ V ) = 1: When rank (Λ M ∩ V ) = 1 all solution points lie on a line. Note that(0 , ∈ Λ M ∩ V , so the line is through the origin, so such set of solutions form a sublattice.Suppose ( a , a ) ∈ Λ M ∩ V is the closest point to the origin, then all other points are a scalar ultiple of ( a , a ). We show that since τ ( x ) = 2 we must have that | a | , | a | ≥ ψ ∗ M M . Supposethat | a | , | a | < ψ ∗ M M < ( ψ ∗ M ) / − ǫ where ǫ >
0. Then it would imply that | a x − a | p < ( ψ ∗ M ) − = (( ψ ∗ M ) / − ǫ ) − − ǫ . However, by our choice of M we have (12), a contradiction, hence | a | , | a | ≥ ψ ∗ M M . By proper-ties of the Euclidean norm we have that λ (Λ M ) ≥ max {| a | , | a |} ≥ ψ ∗ M M .
Noting that any straight line contained in V is of length at most √ M we can deduce thatin the rank M ∩ V ) ≤ √ Mλ + 1 ≤ √ M ψ ( M ) . Taking the maximum of the rank 1 and rank 2 case above we obtain our desired result. (cid:3) Concluding remarks on Theorem 1.2
This article provides sharp bounds on the number of rational points close to almost all p -adic in-tegers. While this result allows us to find simultaneous p -adic Diophantine approximation resultson coordinate hyperplanes, it falls a long way short of providing results for Diophantine approxi-mation sets on curves and manifolds. It is hoped the techniques used in this paper could be usedto find rational points close to curves, we intend to follow this up with a subsequent paper.With regards to the results of this paper, it would be desirable to produce bounds on the number ofrational points close to n -dimensional p -adic integers. In particular for a fixed x = ( x , . . . , x n ) ∈ Z np and n -tuple of approximation functions Ψ = ( ψ , . . . , ψ n ), we would like to find bounds on theset Q n ( x , Ψ , N ) := ( q, q , . . . , q n ) ∈ N × Z n : 0 < q ≤ N, | q i | ≤ N, (cid:12)(cid:12)(cid:12) x i − q i q (cid:12)(cid:12)(cid:12) p < ψ i ( N ) , ≤ i ≤ n . Using the same ideas used in the proof of Theorem 1.2 it can be shown that the set Q n ( x , Ψ , M )can be described by a lattice Λ with basis (11). Intersecting Λ with the the convex body V n :=[0 , M ] × Q ni =1 [ − M, M ] we obtain the set Q n ( x , Ψ , M ). In the cases where rank Λ ∩ V n is full itcan be shown, via Blichfeldt’s Theorem, that Q n ( x , Ψ , M ) ≪ M n +1 n Y i =1 ψ i ( M ) , the expected upper bound for almost all x ∈ Z np . However, for 1 < rank (Λ ∩ V n ) < n issues arisethat cannot be solved by the methods of this chapter. eferences [1] D Allen and S Baker. A general mass transference principle. Selecta Math. (N.S.) , 25(3):Paper No. 39, 38,2019.[2] D Badziahin and Y Bugeaud. On simultaneous rational approximation to a real number and its integral powers,II.
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Email address : [email protected]@york.ac.uk