aa r X i v : . [ m a t h . N T ] N ov ON A COUNTING THEOREM OF SKRIGANOV
NICLAS TECHNAU AND MARTIN WIDMER
Abstract.
We prove a counting theorem concerning the number of latticepoints for the dual lattices of weakly admissible lattices in an inhomogeneouslyexpanding box, which generalises a counting theorem of Skriganov. The errorterm is expressed in terms of a certain function ν (Γ ⊥ , · ) of the dual lattice Γ ⊥ ,and we carefully analyse the relation of this quantity with ν (Γ , · ) . In particular,we show that ν (Γ ⊥ , · ) = ν (Γ , · ) for any unimodular lattice of rank 2, but thatfor higher ranks it is in general not possible to bound one function in termsof the other. Finally, we apply our counting theorem to establish asymptoticsfor the number of Diophantine approximations with bounded denominator asthe denominator bound gets large. Introduction
In the present article, we are mainly concerned with four objectives. Firstly, weprove an explicit version of Skriganov’s celebrated counting result [19, Thm. 6.1] forlattice points of unimodular weakly admissible lattices in homogeneously expandingaligned boxes. Secondly, we use this version to generalise Skriganov’s theorem toinhomogeneously expanding, aligned boxes. Thirdly, we carefully investigate therelation between ν (Γ , · ) (see (1.1) for the definition) and ν (Γ ⊥ , · ) of the dual lattice Γ ⊥ which captures the dependency on the lattice in these error terms. And fourthly,we apply our counting result to count Diophantine approximations.To state our first result, we need to introduce some notation. By writing f ≪ g (or f ≫ g ) for functions f, g , we mean that there is a constant c > such that f ( x ) ≤ cg ( x ) (or cf ( x ) ≥ g ( x ) ) holds for all admissible values of x ; if the impliedconstant depends on certain parameters, then this dependency will be indicatedby an appropriate subscript. Let Γ ⊆ R n be a unimodular lattice, and let Γ ⊥ := { w ∈ R n : h v, w i ∈ Z ∀ v ∈ Γ } be its dual lattice with respect to the standard innerproduct h· , ·i . Let γ n denote the Hermite constant, and for ρ > γ / n set ν (Γ , ρ ) := min (cid:8) | x · · · x n | : x := ( x , . . . , x n ) T ∈ Γ , < k x k < ρ (cid:9) (1.1)where k·k denotes the Euclidean norm. We say Γ is weakly admissible if ν (Γ , ρ ) > for all ρ > γ / n . Note that this happens if and only if Γ has trivial intersection withevery coordinate subspace. Mathematics Subject Classification.
Primary 11P21, 11H06; Secondary 11K60, 22E40,22F30.
Key words and phrases.
Lattice points, counting, Diophantine approximation, inhomogen-eously expanding boxes.The first author was supported by the Austrian Science Fund (FWF): W1230 Doctoral Program“Discrete Mathematics”.
1N A COUNTING THEOREM OF SKRIGANOV 2
Furthermore, let T := diag( t , . . . , t n ) for t i > be the diagonal matrix withdiagonal entries t , . . . , t n , and let y ∈ R n . We set B := T [0 , n + y, and we call such a set an aligned box. Moreover, we define T := (det T ) / n · kT − k = ( t · · · t n ) / n min { t , . . . , t n } ≥ where k · k denotes the operator norm induced by the Euclidean norm. Then, ourgeneralisation of Skriganov’s theorem reads as follows. Theorem 1.
Let n ≥ , let Γ ⊆ R n be a unimodular lattice, and let B ⊆ R n be asabove. Suppose Γ ⊥ is weakly admissible, and ρ > γ / n . Then, | ∩ B ) − vol( B ) | ≪ n ν (cid:0) Γ ⊥ , T ⋆ (cid:1) (cid:18) (vol( B )) − / n √ ρ + R n − ν (Γ ⊥ , R T ) (cid:19) (1.2) where x ⋆ := max { γ n , x } , and R := n + log ρ n ν (Γ ⊥ ,ρT ) . Note that ρ n /ν (Γ ⊥ , ρ ) ≥ n n/ by the inequality between arithmetic and geomet-ric mean. Since T ≥ and γ n ≤ (4 / ( n − / , (1.3)we have (2 R T ) ⋆ = 2 R T , and hence, the far right hand-side in (1.2) is well-defined.The lattice Γ is called admissible if Nm (Γ) := lim ρ →∞ ν (Γ , ρ ) > . It is easy toshow that if Γ is admissible then also Γ ⊥ is admissible (see [18, Lemma 3.1]). Inthis case we can choose ρ = (vol B ) − /n , provided the latter is greater than γ / n ,to recover the following impressive result of Skriganov ([18, Theorem 1.1 (1.11)]) | ∩ B ) − vol( B ) | ≪ n, Nm (Γ ⊥ ) (log(vol( B )) n − . (1.4)However, if Γ is only weakly admissible, then it can happen that Γ ⊥ is not weaklyadmissible; see Example 4. But this is a rather special situation and typically, e.g.,if the entries of A are algebraically independent, see Lemma 3, then Γ = A Z n andits dual are both weakly admissible. This raises the question whether, or underwhich conditions, one can control ν (Γ ⊥ , · ) by ν (Γ , · ) . We have the following resultwhere we use the convention that for an integral domain R the group of all matricesin R n × n with inverse in R n × n is denoted by GL n ( R ) . Proposition 1.
Let
Γ = A Z n , and suppose there exist S, R both in GL n ( Z ) suchthat A T SA = R, and suppose S has exactly one non-zero entry in each column and in each row.Then, we have ν (Γ ⊥ , · ) = ν (Γ , · ) . (1.5)A special case of Proposition 1 shows that ν (Γ ⊥ , · ) = ν (Γ , · ) whenever Γ = A Z n with a symplectic matrix A , in particular, whenever Γ is a unimodular lattice in R . In these cases, one can directly compare Theorem 1 with a recent result [21,Theorem 1.1] of the second author, and we refer to [21] for more on that. On Let us write Sp m ( R ) for the symplectic subgroup of GL m ( R ) and SL n ( R ) for the speciallinear subgroup of GL n ( R ) . The fact Sp ( R ) = SL ( R ) can be checked directly. N A COUNTING THEOREM OF SKRIGANOV 3 the other hand, our next result shows that in general ν (Γ , · ) can decay arbitrarilyquickly even if we control ν (Γ ⊥ , · ) . Theorem 2.
Let n ≥ , and let ψ : (0 , ∞ ) → (0 , be non-increasing. Then,there exists a unimodular, weakly admissible lattice Γ ⊆ R n , and a sequence { ρ l } ⊆ ( γ / n , ∞ ) tending to ∞ , as l → ∞ , such that ν (Γ ⊥ , ρ ) ≫ ρ − n , and ν (Γ , ρ l ) ≤ ψ ( ρ l ) for all l ∈ N = { , , , . . . } and for all ρ > γ / n . In the case where exactly one of the functions ν (Γ , · ) , and ν (Γ ⊥ , · ) is controllablewhile the other one decays very quickly either Theorem 1 or [21, Theorem 1.1]provides a reasonable error term, but certainly not both. This highlights the com-plementary aspects of Theorem 1, and [21, Theorem 1.1]. Theorem 2 is deeper thanProposition 1, and relies on a recent result of Beresnevich about the distribution ofbadly approximable vectors on manifolds.Next, we apply Theorem 1 to deduce counting results for Diophantine approxim-ations. We start with a bit of historical background on this, and related problems.Let α ∈ R , let ι : [1 , ∞ ) → (0 , be a positive decreasing function, and let N locα ( ι, t ) be the number of integer pairs ( p, q ) satisfying | p + qα | < ι ( q ) , ≤ q ≤ t . In a seriesof papers, starting in 1959, Erdős [13], Schmidt [16, 17], Lang [9, 14, 15], Adams[1, 2, 3, 4, 5, 6, 7, 8], Sweet [20], and others, considered the problem of finding theasymptotics for N locα ( ι, t ) as t gets large.Schmidt [16] has shown that for almost every α ∈ R the asymptotics are given bythe volume of the corresponding subset of R , provided the latter tends to infinity.This is false for quadratic α ; there with ι ( q ) = 1 /q the volume is t ) + O (1) ,and by Lang’s result N locα (1 /q, t ) ∼ c α log( t ) but Adams [5] has shown that c α = 2 .Opposed to the above “localised” setting, where the bound on | p + qα | is expressedas a function of q , we consider the “non-localised” (sometimes called “uniform”)situation, where the bound is expressed as a function of t . Furthermore, we shallconsider the more general asymmetric inhomogeneous setting. Let α ∈ (0 , beirrational, ε, t ∈ (0 , ∞ ) , and let y ∈ R . We define the counting function N α,y ( ε, t ) = (cid:26) ( p, q ) ∈ Z × N : 0 ≤ p + qα − y ≤ ε, ≤ q ≤ t (cid:27) . (1.6)If the underlying set is not too stretched, then N α,y ( ε, t ) is roughly the volume εt of the set in which we are counting lattice points. If we let ε = ε ( t ) be a functionof t with t = o ( tε ) we have, by simple standard estimates, N α,y ( ε, t ) ∼ εt (1.7)for any pair ( α, y ) ∈ ((0 , \ Q ) × R whatsoever. To get non-trivial estimates forour counting function, we need information on the Diophantine properties of α . Let φ : (0 , ∞ ) → (0 , be a non-increasing function such that q (cid:12)(cid:12) p + qα (cid:12)(cid:12) ≥ φ ( q ) (1.8) Here “almost every” refers always to the Lebesgue measure.
N A COUNTING THEOREM OF SKRIGANOV 4 holds for all ( p, q ) ∈ Z × N . Then [21, Theorem 1.1] implies that | N α,y ( ε, t ) − εt | ≪ α s εtφ ( t ) . (1.9)Hence, unlike in the localised setting, for badly approximable α the asymptoticsare given by the volume as long as the volume tends to infinity.Our next result significantly improves the error term in (1.9), provided α is“sufficiently” badly approximable, i.e., provided φ ( t ) decays slowly enough. Weassume that εt > < ε < √ α. (1.10) Corollary 1.
Put E := εtφ (4 t √ εt ) , and E ′ := 168 √ εt E . Then, we have | N α,y ( ε, t ) − εt | ≪ α log Eφ ( E ′ ) . (1.11)In particular, if α is badly approximable then | N α,y ( ε, t ) − εt | ≪ α log( εt ) . (1.12)2. An explicit version of Skriganov’s counting theorem
Let Γ ⊆ R n be a lattice, and let λ i (Γ) denote the i -th successive minimum of Γ with respect to the Euclidean norm ( ≤ i ≤ n ). For r > we introduce a specialset of diagonal matrices ∆ r := (cid:8) δ := diag(2 m , . . . , m n ) : m = ( m , . . . , m n ) T ∈ Z n , k m k < r, det δ = 1 (cid:9) , and we put S (Γ , r ) := X δ ∈ ∆ r ( λ ( δ Γ)) − n . Now we can state Skriganov’s result. In fact, his result is more general, and appliesto any convex, compact polyhedron. On the other hand, the dependency on B and Γ in the error term is not explicitly stated in his counting result [19, Thm. 6.1]. Bycarefully following his reasoning, see Remark 1 below, we find the following explicitversion of his result. Recall that γ n denotes the Hermite constant. Theorem 3. [Skriganov, 1998] Let n ≥ be an integer, let Γ ⊆ R n be a unimodularlattice, and let B ⊆ R n be an aligned box of volume . Suppose Γ ⊥ is weaklyadmissible, and ρ > γ / n . Then, for t > , | ∩ tB ) − t n | ≪ n ( | ∂B | λ n (Γ)) n · ( t n − ρ − / + S (Γ ⊥ , r )) (2.1) where r := n + log ρ n ν (Γ ⊥ ,ρ ) , and | ∂B | denotes the surface area of B .Remark . The references and notation in this remark are the same as in [19]. Put O := tB , fix a mollifier ω as in (11.3), and denote by ˜ χ ( O , · ) the Fourier transformof the characteristic function χ ( O , · ) of O . Skriganov applies Lemma 11.1 to theerror term R ( O , Γ) := sup X ∈ R n | O + X ) ∩ Γ) − vol( O ) | to estimate it by R ( O , Γ) ≤ vol( O + τ ) − vol( O − τ ) + sup X ∈ R n (cid:0)(cid:12)(cid:12) R + τ ( O , X ) (cid:12)(cid:12) + (cid:12)(cid:12) R − τ ( O , X ) (cid:12)(cid:12)(cid:1) N A COUNTING THEOREM OF SKRIGANOV 5 where O ± τ is a τ -coapproximation of O , and R ± τ are the Fourier series R ± τ ( O , X ) := X γ ∈ Γ ⊥ \{ } ˜ χ ( O ± τ , γ )˜ ω ( τ γ ) e − πi h γ,X i defined in (11.5) where ˜ ω denotes the Fourier transform of ω . Observe that | ∂B | ≥ ,and that without loss of generality B is centred at the origin, i.e., y = − ( t , . . . , t n ) T .Hence, we can choose O ± τ := ( t ± | ∂B | τ ) B with < τ < , and thus vol( O + τ ) − vol( O − τ ) ≪ n | ∂B | n t n − τ. As noted in (6.6), since B is an aligned box, the average S (Γ f , · ) simplifies to S (Γ ⊥ , · ) , and ν (Γ ⊥ f , · ) = ν (Γ ⊥ , · ) for each flag of faces f of B .Now R ± τ is decomposed via (12.7) into partial sums A ± τ,ρ plus remainder terms B ± τ,ρ which are defined in (12.8) and (12.9), respectively. Let ω denote the Fouriertransform of ω (cf. p. 57). Due to (12.12), there is a constant c = c ( ω , ω ) ,independent of Γ , t, ρ, τ , such that max X ∈ R n A ± τ,ρ ( O , X ) ≤ cS (Γ ⊥ , r ) where we may choose r to be r := n + log ρ n ν (Γ ⊥ , ρ ) . Hence, c depends in fact only on the (fixed) mollifier ω . Furthermore, B ± τ,ρ ( O , X ) is estimated in (12.14) by max X ∈ R n (cid:12)(cid:12) B ± τ,ρ ( O , X ) (cid:12)(cid:12) ≤ c A π | ∂B | t n − τ − A X γ ∈ Γ ⊥k γ k > ρ k γ k − A − where A > n . Note that for
R > (cid:8) γ ∈ Γ ⊥ : k γ k < R (cid:9) ≪ n ( R/λ (Γ ⊥ ) + 1) n . This in turn implies that for k ∈ N we have (cid:8) γ ∈ Γ ⊥ : 2 k ≤ k γ k < k +1 (cid:9) ≪ n (2 k +1 /λ (Γ ⊥ )) n . Using dyadic summation, and Mahler’s relations ≤ λ i (Γ ⊥ ) λ n +1 − i (Γ) ≤ n ! ( i = 1 , . . . , n ) (2.2)yields X γ ∈ Γ ⊥k γ k > ρ k γ k − A − ≪ n X k> j log(8 − ρ )log 2 k ( k +1) n λ − n (Γ ⊥ ) · − ( A +1) k ≪ n λ nn (Γ) ρ n − A − . Hence, (cid:12)(cid:12) R ± τ ( O , X ) (cid:12)(cid:12) ≪ n cS (Γ ⊥ , r ) + c A | ∂B | t n − τ − A λ nn (Γ) ρ n − A − . Given a compact region
O ⊆ R n and a real number τ > , compact regions O ± τ are called τ -coapproximations to O , if O − τ ⊆ O ⊆ O + τ and dist( ∂ O , ∂ O ± τ ) ≥ τ are satisfied. Conceivably, we should mention a typo regarding the definition of r f in (6.5): r f is to be takenas in (12.13). In (12.13) κ n denotes τ n from Lemma 10.1, which was defined in (7.4) as two timesthe diameter of the Dirichlet-Voronoi region of the lattice M defined in (3.3). It is easy to seethat τ n < n . N A COUNTING THEOREM OF SKRIGANOV 6
Specialising A := 2 n − implies R ( O , Γ) ≪ n | ∂B | n t n − τ + S (Γ ⊥ , r ) + | ∂B | n t n − τ − n λ nn (Γ) ρ − n ≪ n ( | ∂B | λ n (Γ)) n ( t n − τ + S (Γ ⊥ , r ) + t n − τ − n ρ − n ) where in the last inequality we used the obvious fact | ∂B | ≥ . Finally, choosing τ := ρ − / gives the required estimate.For proving Theorem 1, we want to exploit Theorem 3. To this end let t :=(det T ) / n , and let U := t T − . (2.3)Thus, ∩ B ) = U Γ ∩ U ( T [0 , n + y )) = ∩ t ([0 , n + T − ( y ))) where Λ := U Γ . Moreover, we conclude by Theorem 3 that | ∩ B ) − vol( B ) | ≪ n λ nn (Λ) (cid:18) t n − √ ρ + S (Λ ⊥ , r ) (cid:19) . (2.4)For controlling the quantities on the right hand side in terms of Γ , t , ρ , and ν (Γ ⊥ , · ) ,we need two lemmata. We will frequently use the fact that if Γ = A Z n is unimodularthen Γ ⊥ = ( A − ) T Z n . As usual, we let SL n ( R ) denote the group of all R n × n matrices with determinant . Lemma 1.
Let D := diag( d , . . . , d n ) be in SL n ( R ) , and ρ > γ / n . Then, ν (( D Γ) ⊥ , ρ ) ≥ ν (Γ ⊥ , k D k ρ ) , (2.5) and λ n ( D Γ) ≫ n ν (Γ , (cid:13)(cid:13) D − (cid:13)(cid:13) ⋆ ) . (2.6) Proof.
For v := ( v , . . . , v n ) T ∈ R n define Nm( v ) := | v · · · v n | . We remark that ν (( D Γ) ⊥ , ρ ) = ν ( D − Γ ⊥ , ρ )= min (cid:8) Nm( D − v ) : v ∈ Γ ⊥ , < (cid:13)(cid:13) D − v (cid:13)(cid:13) < ρ (cid:9) = min (cid:8) Nm ( v ) : v ∈ Γ ⊥ , < (cid:13)(cid:13) D − v (cid:13)(cid:13) < ρ (cid:9) . If k D − v k < ρ , then k v k < k D k ρ . Thus, (2.5) follows. Now let Q > , and v ∈ Γ with < k v k ≤ Q . By the inequality of arithmetic and geometric mean, wehave k Dv k n ≥ n n / · Nm( Dv ) ≫ n ν (Γ , Q ⋆ ) . Now suppose k v k > Q . Since k v k = (cid:13)(cid:13) D − Dv (cid:13)(cid:13) ≤ (cid:13)(cid:13) D − (cid:13)(cid:13) k Dv k , we concludethat k Dv k > (cid:13)(cid:13) D − (cid:13)(cid:13) − Q. Hence, we have k Dv k ≫ n min (cid:8) ( ν (Γ , Q ⋆ )) / n , (cid:13)(cid:13) D − (cid:13)(cid:13) − Q (cid:9) . Specialising Q := (cid:13)(cid:13) D − (cid:13)(cid:13) , and noticing that by the inequality of arithmetic andgeometric mean, ν (Γ , γ n ) ≪ n , we get (2.6). (cid:3) N A COUNTING THEOREM OF SKRIGANOV 7
Lemma 2.
Let U be as in (2.3), and let s ≥ . Then, we have S (Λ ⊥ , s ) ≪ n s n − ν (Γ ⊥ , (2 s k U k ) ⋆ ) . Proof.
Since Λ ⊥ = U − Γ ⊥ , we conclude by (2.6) that S (Λ ⊥ , s ) = X δ ∈ ∆ s λ n ( δU − Γ ⊥ ) ≪ n X δ ∈ ∆ s ν (Γ ⊥ , k U δ − k ⋆ ) . Since s ≪ n s n − , and since ν (Γ ⊥ , · ) is non-increasing, we get S (Λ ⊥ , s ) ≪ n s n − ν (Γ ⊥ , (2 s k U k ) ⋆ ) . (cid:3) Now we can give the proof of Theorem 1.
Proof of Theorem 1.
By (2.5), we conclude r = n + log ρ n ν (Λ ⊥ , ρ ) ≤ n + log ρ n ν (Γ ⊥ , k U k ρ ) = R Since ν (Λ ⊥ , · ) is non-increasing, and since (2 R k U k ) ⋆ = 2 R k U k Lemma 2 yields S (Λ ⊥ , r ) ≪ n R n − ν (Γ ⊥ , R k U k ) . (2.7)By using Mahler’s relation (2.2) and Lemma 1, we obtain λ nn (Λ) ≪ n λ n ( U − Γ ⊥ ) ≪ n ν (Γ ⊥ , k U k ⋆ ) . (2.8)Taking (2.7) and (2.8) in (2.4) into account, it follows that | ∩ B ) − vol( B ) | ≪ n ν (cid:0) Γ ⊥ , k U k ⋆ (cid:1) (cid:18) t n − √ ρ + R n − ν (Γ ⊥ , R k U k ) (cid:19) which is (1.2). (cid:3) Comparing ν (Γ , · ) and ν (Γ ⊥ , · ) A natural question is whether one can state Theorem 1 in a way that is intrinsicin Γ , i.e. expressing ν (Γ ⊥ , · ) in terms of ν (Γ , · ) . However, for n > there are weaklyadmissible lattices Γ ⊆ R n such that Γ ⊥ is not weakly admissible as the followingexample shows. Example 4.
Let n ≥ , and let A ′ ∈ GL n − ( R ) be such that the elements of eachrow of A ′ are Q -linearly independent. Choose real x , . . . , x n − , y outside of the Q -span of the entries of A ′ , and suppose y = x n − . Let x = ( x , . . . , x n − ) T andlet r n − be the last row of A ′ . Then, the matrix A := (cid:18) A ′ xr n − y (cid:19) satisfies(i) A ∈ GL n ( R ) , and(ii) the elements in each row of A are Q -linearly independent. N A COUNTING THEOREM OF SKRIGANOV 8
The second assertion is clear and for the first suppose a linear combination of therows vanishes. Using that the rows of A ′ are linearly independent over R and that y = x n − , the first claim follows at once. We now let A be the matrix we get from A by swapping the first and the last row, and scaling each entry with | det A | − /n .Clearly, (i) and (ii) remain valid for A , and the ( n, n ) -minor of A vanishes. Weconclude that Γ := A Z n is a unimodular, and weakly admissible lattice; moreover,Cramer’s rule implies that ( A − ) T = ⋆ ⋆ . . . ⋆⋆ . . . . . . ...... . . . ⋆ ⋆⋆ . . . ⋆ where an asterisk denotes some arbitrary real number, possibly a different numbereach time. Hence, Γ ⊥ contains a non-zero lattice point with a zero coordinate, andthus is not weakly admissible. Keeping Example 4 in mind, we now concern ourselves with finding large sub-classes of lattices Γ ⊆ R n such that(1) Γ and Γ ⊥ are both weakly admissible,(2) ν (Γ ⊥ , · ) = ν (Γ , · ) .It is easy to see that the first item holds for almost all lattices in the sense ofthe Haar-measure on the space L n = SL n ( R ) / SL n ( Z ) of unimodular lattices in R n .Moreover, we have the following criterion. Lemma 3.
Suppose A ∈ SL n ( R ) , and suppose that the entries of A are algebraicallyindependent (over Q ). Then, Γ := A Z n and Γ ⊥ are both weakly admissible.Proof. First note that if K is a field and X , . . . , X N are algebraically independentover K , then any non-empty collection of pairwise distinct monomials X a · · · X a N N is linearly independent over K . Next note that by Cramer’s rule, each entry of ( A − ) T is a sum of pairwise distinct monomials (up to sign) in the entries of A ,and none of these monomials occurs in more than one entry of ( A − ) T . This showsthat the entries of ( A − ) T are linearly independent over Q , in particular, the entriesof any fixed row of ( A − ) T are linearly independent over Q . Thus, Γ ⊥ is weaklyadmissible. (cid:3) Next, we prove Proposition 1. Notice that S and S − are, up to signs of theentries, permutation matrices, and thus for every w ∈ R n Nm ( w ) = Nm ( Sw ) = Nm ( S − w ) , (3.1) k w k = k Sw k = (cid:13)(cid:13) S − w (cid:13)(cid:13) . (3.2)Now let Aw be an arbitrary lattice point in Γ = A Z n . Then, since R ∈ Z n × n , we get ( A − ) T Rw ∈ Γ ⊥ . Since by hypothesis A = S − (( A − ) T R ) , we conclude from (3.1)that Nm ( Aw ) = Nm (( A − ) T Rw ) , and from (3.2) that k Aw k = (cid:13)(cid:13) ( A − ) T Rw (cid:13)(cid:13) .This shows that ν (Γ ⊥ , · ) ≤ ν (Γ , · ) .Similarly, if ( A − ) T w ∈ Γ ⊥ then, since R − ∈ Z n × n , we find that AR − w ∈ Γ ,and using that ( A − ) T = SAR − we conclude as above that ν (Γ , · ) ≤ ν (Γ ⊥ , · ) .This proves Proposition 1. N A COUNTING THEOREM OF SKRIGANOV 9
Remark . Let I m := diag(1 , . . . , be the identity matrix, and m the null matrixin R m × m . Specialising S = R = (cid:18) m I m − I m m (cid:19) in Proposition 1, we conclude that if Γ = A Z n with a symplectic matrix A , then ν (Γ ⊥ , · ) = ν (Γ , · ) . (3.3)Moreover, it is easy to see that Sp ( R ) = SL ( R ) , and hence (3.3) holds for anyunimodular lattice Γ ⊆ R .Next, we prove Theorem 2. Recall that α := ( α , . . . , α n ) T ∈ R n is called badlyapproximable, if there is a constant C = C ( α ) > such that for any integer q ≥ the inequality max {k qα k , . . . , k qα n k} ≥ Cq / n (3.4)holds where k·k denotes the distance to the nearest integer. By a well-knowntransference principle, cf. [12], assertion (3.4) is equivalent to saying that for allnon-zero vectors q := ( q , . . . , q n ) T ∈ Z n the inequality kh α, q ik ≥ ˜ C k q k n (3.5)holds where ˜ C = ˜ C ( α ) > is a constant. Let Bad ( n ) denote the set of all badlyapproximable vectors in R n . The crucial step for constructing matrices generatingthe lattices announced in Theorem 2 is done by the following lemma. Lemma 4.
Let n ≥ be an integer. Fix algebraically independent real numbers c i,j where i, j = 1 , . . . , n and i = j . Then, there exist λ , . . . , λ n ∈ R such that theentries of each row of A := λ c , . . . c ,n c , λ . . . ...... . . . . . . c n − ,n c n, . . . c n,n − λ n (3.6) are algebraically independent, A is invertible, and each row-vector of ( A − ) T isbadly approximable. For proving this lemma, we shall use the following special case of a recent The-orem of Beresnevich concerning badly approximable vectors. We say that the map F := ( f , . . . , f n ) T : B → R n , where B ( R m is a non-empty ball and m, n ∈ N , isnon-degenerate, if , f , . . . , f n are linearly independent functions (over R ). Theorem 5 ([10, Thm. 1]) . Let n, m, k be positive integers. For each j = 1 , . . . , k suppose that F j : B → R n is a non-degenerate, analytic map defined on a non-emptyball B ( R m . Then, dim Haus k \ j =1 F − j ( Bad ( n )) = m. N A COUNTING THEOREM OF SKRIGANOV 10
Proof of Lemma 4.
We work in two steps. First, we set the scene to make use ofTheorem 5.(i) Let M ∈ R n × n , and denote by ( M ) i,j the entry in the i -th row and j -thcolumn of M . Moreover, we define a map ˜ F : R n → R n × n by λ := ( λ , . . . , λ n ) T λ c , . . . c ,n c , λ . . . ...... . . . . . . c n − ,n c ,n . . . c n,n − λ n . On a sufficiently small non-empty ball B ( R n , centred at the origin, ˜ F ( λ ) isinvertible for every λ ∈ B . On this ball B , we define F j , for j = 1 , . . . , n , bymapping λ to the j -th row of ( (cid:0) ˜ F ( λ ) (cid:1) − ) T . We claim that F j is a non-degenerate,and analytic map. By Cramer’s rule, every entry of (( ˜ F ( λ )) − ) T is the quotientof polynomials in λ , . . . , λ n whereas the polynomial in the denominator does notvanish on B . Hence, each F j is an analytic function. Now we show that F isnon-degenerate, the argument for the other F j being similar. The j -th componentof F is ( (cid:0) ˜ F ( λ ) (cid:1) − ) j, and, using Cramer’s rule, is hence of the shape (det ˜ F ( λ )) − R j + ( − j n Y k =2 , k = j λ k ! where the polynomial R j ∈ R [ λ , . . . , λ n ] is of (total) degree < n − , if j = 1 ,and of (total) degree < n − , if j = 2 , . . . , n . Therefore, if a linear combination k + P nj =1 k j (( ˜ F ( λ )) − ) j, with scalars k , . . . , k n ∈ R equals the zero-function : B → R , then = k · (det ˜ F ( λ )) + n X j =1 k j ( − j n Y k =2 , k = j λ k + n X j =1 k j R j . Comparing coefficients, we conclude that k = 0 and thereafter k = k = · · · = k n = 0 . Hence, F is non-degenerate.(ii) By part (i), Theorem 5 implies that the set M of all λ ∈ B such that F ( λ ) , . . . , F n ( λ ) are all badly approximable, has full Hausdorff dimension. Moreover,we claim that there is a set M (1) ⊆ M of full Hausdorff dimension such that forevery λ ∈ M (1) the entries of the first row of ˜ F ( λ ) are algebraically independent.Let M be the subset of M of all elements λ := ( λ , . . . , λ n ) T ∈ M satisfying that { λ , c ,j : j = 2 , . . . , n } is algebraically dependent ; observe that the possible val-ues for λ are countable, since Z [ c , , . . . , c ,n , x ] is countable and every complex,non-zero, univariate polynomial has only finitely many roots. Therefore, M iscontained in a countable union of hyperplanes. It is well-known that if a sequence To see this, it suffices to show det ˜ F ((0 , . . . , T ) = 0 . However, by the Leibniz formula, det ˜ F (0 , . . . ,
0) = X σ sgn( σ ) n Y i =1 c i,σ ( i ) where the sum runs through all fixpoint-free permutations of { , . . . , n } . Since { c i,j : i, j = 1 , . . . , n, i = j } is algebraically independent, the evaluation of the polynomial on theright hand side above cannot vanish, cf. proof of Lemma 3. N A COUNTING THEOREM OF SKRIGANOV 11 of sets { E i } ⊆ R n is given, then dim Haus S i ≥ E i = sup i ≥ { dim Haus E i } , cf. [11, p.65]. Consequently, n = dim Haus M = max { dim Haus ( M \ M ) , dim Haus M } = dim Haus ( M \ M ) , and we define M (1) := M \ M . Using the same argument, we conclude that thereis a set M (2) ⊆ M (1) of full Hausdorff dimension such that each of the first tworows of ˜ F ( λ ) has algebraically independent entries for every λ ∈ M (2) . Iteratingthis construction, we infer that there is a subset M ( n ) ⊆ M ( n − ⊆ . . . ⊆ M of fullHausdorff dimension such that for every λ ∈ M ( n ) each row of the matrix A := ˜ F ( λ ) has algebraically independent entries, and ( A − ) T has badly approximable rowvectors. Moreover, λ ∈ M ( n ) ⊆ B implies that A is invertible. (cid:3) We also need the following easy fact whose proof is left as an exercise.
Lemma 5.
Let m ∈ N , and let α ∈ R be transcendental. Then, there are realnumbers β , . . . , β m such that β , αβ , β , . . . , β m are algebraically independent.Proof of Theorem 2. First, we set ˜ ψ ( x ) = ψ ( x ) such that for every c > and x ≥ c we have ˜ ψ ( x ) ≤ ψ ( cx ) . We may assume that ˜ ψ ( q ) ≪ exp( − q ) . By writing down asuitable decimal expansion, we conclude that there exists a number α ∈ (0 , suchthat (cid:12)(cid:12)(cid:12) α − pq (cid:12)(cid:12)(cid:12) < ˜ ψ ( q ) q n +1 (3.7)has infinitely many coprime integer solutions p, q ∈ Z ; observe that such an α is necessarily transcendental. We apply Lemma 5 with m = n − n and we set c , := β , c , := αβ , and we choose exactly one value β k ( k ≥ ) for each of theremaining c i,j ( i = j ). Thus, the real numbers c i,j are algebraically independent.We use Lemma 4 with these specifications to find A as in (3.6). For l ∈ N let p l , q l denote distinct solutions to (3.7), and put v l := (0 , − p l , q l , , . . . , T ∈ Z n . Set ˜ A := | det A | − /n A , and let us consider the unimodular, weakly admissible lattice Γ := ˜ A Z n . Then, the first coordinate of ˜ Av l equals | det A | − /n |− p l c , + q l c , | = | det A | − /n | c , | | q l α − p l | ≪ A ˜ ψ ( q l ) q nl . Since α ∈ (0 , , we may assume, by choosing l large enough, that p l ≤ q l . Hence,the j -th coordinate for j = 2 , . . . , n of ˜ Av l is ≪ A q l . Thus, for l sufficiently large, Nm( ˜ Av l ) ≪ A ˜ ψ ( q l ) q nl · q n − l = ˜ ψ ( q l ) q l ≤ ψ (2 k ˜ A k q l ) q l ≤ ψ ( k ˜ Av l k ) q l . Choosing ρ l = k ˜ Av l k , we conclude that ν (Γ , ρ l ) ≤ ψ ( ρ l ) for all l sufficiently large.Because the rows of ( A − ) T are badly approximable vectors by construction, Γ ⊥ is weakly admissible. Moreover, by (3.5), we conclude that Nm(( A − ) T v ) ≫ A k v k − n for every non-zero v ∈ Z n . Also note that (cid:13)(cid:13) ( A − ) T v (cid:13)(cid:13) < ρ implies k v k < k A T k ρ .This implies that ν (Γ ⊥ , ρ ) ≫ A ρ − n . Hence, Γ has the desired properties. (cid:3) N A COUNTING THEOREM OF SKRIGANOV 12 An Application - Proof of Corollary 1
Throughout this section we fix the unimodular lattice
Γ = A Z where A := 1 √ α (cid:18) α α (cid:19) , and we consider the aligned box B := 1 √ α (cid:0)(cid:2) y, y + ε (cid:3) × (cid:2) y, y + αt (cid:3)(cid:1) . (4.1)Then, the following relation holds B ∩ Γ) = (cid:26) ( p, q ) ∈ Z : 0 ≤ p + αq − y ≤ ε, ≤ p + 2 αq − y ≤ αt (cid:27) . Because of (1.10), we conclude that | N α,y ( ε, t ) − B ∩ Γ) | ≪ α . (4.2)In order to use Theorem 1, we need to control the characteristic quantity ν (Γ , · ) of the lattice Γ . This is where the Diophantine properties of α come into play. Lemma 6.
Let φ be as in (1.8), and suppose ρ > γ / . Then, we have ν (Γ ⊥ , ρ ) = ν (Γ , ρ ) ≥ φ (4 ρ/ √ α )4 . Proof.
The claimed equality follows immediately from Proposition 1, and the re-mark thereafter. A vector v ∈ Γ is of the shape v = 1 √ α (cid:18) zz ′ (cid:19) where z := p + qα , z ′ := z + qα , and p, q denote integers. Assume that k v k ∈ (0 , ρ ) .Observe that q = 0 implies Nm( v ) ≥ > − φ (4 ρ/ √ α ) . Therefore, we may assume q = 0 . Since z ′ − z = qα , one of the numbers | z | , | z ′ | is at least α | q | , and both arebounded from below by | q | φ (2 | q | ) . Hence, Nm ( v ) ≥ α | q | √ α · φ (2 | q | )2 | q |√ α ≥ φ (4 ρ/ √ α )4 where in the last step we used that √ α | q | ≤ √ α min {| z | , | z ′ |} ≤ k v k < ρ. (cid:3) Proof of Corollary 1.
Let B be given by (4.1). Thus, B has sidelengths t = α − / ε ,and t = √ αt . By (1.3) and (1.10), we are entitled to take ρ := εt > γ / in Theorem1. Moreover, (1.10) implies t < < t , and thus T = r α tε > √ εt > > γ . Hence, T ⋆ = T . By combining relation (4.2) and Theorem 1 with these specifica-tions, it follows that | N α,y ( ε, t ) − εt | ≪ α ν (Γ ⊥ , T ) (cid:18) Rν (Γ ⊥ , R T ) (cid:19) . (4.3) N A COUNTING THEOREM OF SKRIGANOV 13
By Lemma 6, the right hand side above is ≪ R ( φ (4 T / √ α ) φ (2 R +2 T / √ α )) − . Thefirst factor in the round brackets is larger than the second one, since φ is non-increasing. Hence, we conclude that the right hand-side of (4.3) is bounded by ≪ R ( φ (2 R +2 T / √ α )) − . (4.4)Furthermore, Lemma 6 yields R ≤ εt ) φ (4 t √ εt ) ≪ log εtφ (4 t √ εt ) . (4.5)By using the first estimate from (4.5), we get R ≤ (cid:18) εt ) φ (4 t √ εt ) (cid:19) log 2 < ( εt ) φ (4 t √ εt ) . Hence, (4.4) is bounded from above by ≪ log εtφ (4 t √ εt ) φ (cid:16) εt ) φ (4 t √ εt ) q tε (cid:17) ≤ log Eφ ( E ′ ) . This completes the proof of Corollary 1. (cid:3) acknowledgements
We would like to thank Carsten Elsner for sending us a preprint of his work.
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