WEAK ADMISSIBILITY, PRIMITIVITY, O-MINIMALITY, AND DIOPHANTINE APPROXIMATION

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1 WEAK ADMISSIBILITY PRIMITIVITY O-MINIMALITY AND DIOPHANTINE APPROXIMATION MARTIN WIDMER Contents 1. Introduction 1 2. Generalisation of weak admissibility and statement of the results Generalised weak admissibility Generalised aligned boxes Main results 5 3. Basic counting principle 7 4. Proof of Theorem Preparations for the Möbius inversion Proof of Theorem Lower bounds for the error term F κm - Families via o-minimality 16 Acknowledgements 17 References 17 Abstract. We generalise M. M. Skriganov s notion of weak admissibility for lattices to include standard lattices occurring in Diophantine approximation and algebraic number theory and we prove estimates for the number of lattice points in sets such as aligned boxes. Our result improves on Skriganov s celebrated counting result if the box is sufficiently distorted the lattice is not admissible and e.g. symplectic or orthogonal. We establish a criterion under which our error term is sharp and we provide examples in dimensions 2 and 3 using continued fractions. We also establish a similar counting result for primitive lattice points and apply the latter to the classical problem of Diophantine approximation with primitive points as studied by Chalk Erdős and others. Finally we use o-minimality to describe large classes of sets to which our counting results apply. 1. Introduction In this article we generalise Skriganov s notion of weak) admissibility for lattices to include standard lattices occurring in Diophantine approximation and algebraic number theory e.g. ideal lattices) and we prove a sharp estimate for the number of lattice points in sets such as aligned boxes. Our result applies when the lattice is weakly admissible whereas Skriganov s result requires the dual lattice to be weakly admissible in his stronger sense). If the lattice is symplectic or orthogonal 1 and weakly admissible then both results apply and our error term is better provided the lattice is not admissible and Date: December Mathematics Subject Classification. Primary 11H06 11P21 11K60 11J20; Secondary 03C64 22F30. Key words and phrases. Weakly admissible lattice points Diophantine approximation counting coprime primitive o-minimality flows homogeneous spaces. 1 The lattice Λ is symplectic or orthogonal) if Λ = AZ N for some symplectic or orthogonal) A GL N R). 1

2 2 MARTIN WIDMER the box is sufficiently distorted. Our error term also has a good dependence on the geometry of the lattice which allows us to apply a Möbius inversion to get a similar estimate for primitive lattice points. The motivation for this comes from a classical Diophantine approximation result [4] due to Chalk and Erdős from 1959 for numbers; it appears that our result is the first one in higher dimensions. We also make modest progress on a conjecture of Dani Laurent and Nogueira [5 10] on an inhomogeneous Khintchine Groshev type result for primitive points. Finally we use o-minimality a notion from model theory to describe large classes of sets to which our counting results apply. The usage of o-minimality to asymptotically count lattice points has been initiated by Barroero and the author [3] and [3 Theorem 1.3] has already found various applications see e.g. [ ]). Here we further develop this idea but we use o-minimality in a different way. Next we shall state the simplest special case of Theorem 2.1 and compare it to Skriganov s result [17 Theorem 6.1] more precisely Technau and the authors generalisation [19 Theorem 1] from uniformly to non-uniformly scaled boxes). Let Γ R N. Following Skriganov we define νγ ϱ) := inf{ x 1 x N 1/N ; x Γ\{0} x < ϱ} and we say a lattice Λ in R N is weakly admissible if νλ ϱ) > 0 for all ϱ > 0 and admissible if lim ϱ νλ ϱ) > 0. Let Z be a translate of the box [ 1 1 ] [ N N ] and write max for the maximal i and for their geometric mean. We set E Λ Z ) := #Λ Z ) VolZ / det Λ. Theorem 1.1. Suppose Λ is a weakly admissible lattice in R N. Then we have E Λ Z ) N inf 0<B max νλ B) + ) N 1 max. B Suppose Λ is unimodular. Skriganov [17 Theorem 6.1] proved error estimates for uniformly scaled aligned boxes and more generally certain polyhedrons) provided the dual lattice Λ with respect to the standard inner product) is weakly admissible see also [ ) Theorem 1.1] for a precursor of this result for admissible lattices). As shown in [19 Theorem 1] his method also leads to results for non-uniformly scaled aligned boxes provided Λ is weakly admissible) of the form 2 1.1) E Λ Z ) N 1 νλ / min ) ) N inf ϱ>γ 1/2 N N 1 ϱ + r N 1 νλ 2 r / min ) N where γ N denotes the Hermite constant r = N 2 + N logϱ/νλ ϱ/ min )) and / min ) = max{/ min γ N }. If Λ is admissible which implies that Λ is admissible) then Skriganov s bound becomes Λ log ) N 1 which conjecturally is sharp. Let us now suppose that Λ is weakly admissible but not admissible. Technau and the author [19 Theorem 2] have shown that in general even if Λ and Λ are both weakly admissible there is no way to bound νλ ) in terms of νλ ). This indicates the complementary aspect of Theorem 1.1 and 1.1). However if Λ = AZ N with e.g. a symplectic or orthogonal matrix A then νλ ) = νλ ) by [19 Proposition 1] and we can directly compare our result with Skriganov s; note also that for N = 2 every unimodular lattice is symplectic cf. [19 Remark after Proposition 1]). Using that / min max /) 1/N 1) =: and that r N logνλ )) we find the following crude lower bound 3 for the right hand-side of 1.1) νλ )νλ νλ ) N log 2 ) ) N 1.2). 2 In the above setting our definition of ν ) is the N-th root of Skriganov s and the one in [19]. 3 We are only interested in sufficiently distorted boxes and so we can assume > γ 1/2 N. )

3 WEAK ADMISSIBILITY PRIMITIVITY O-MINIMALITY AND DIOPHANTINE APPROXIMATION3 Choosing B = max / = N 1 we see that the error term in Theorem 1.1 is bounded from above by 1.3) N N 1 νλ N 1 ) N 1). In particular if N = 2 then our error term is better whenever νλ max /) 3 is larger than a certain multiple of VolZ ) 1/2 so if the box is sufficiently distorted in terms of νλ ) and the volume of the box note that for νλ max /) 1 = o) as tends to infinity we still get asymptotics). Also for arbitrary N our error term is better when the box is sufficiently distorted in terms of νλ ) and the volume of the box and provided νλ ϱ) decays faster than ϱ 1/ log 2 or sufficiently slowly e.g. like a negative power of log ϱ. The latter happens for almost every unimodular lattice cf. [17 Lemma 4.5]) and with Λ = AZ N also for almost every 4 matrix A SO N R) cf. [17 Lemma 4.3]) and as mentioned before for these Λ we also have νλ ) = νλ ). Another significant difference between our and Skriganov s error term concerns the dependence on the lattice. If we replace Z by k 1 Z or equivalently replace Λ by kλ and fix Z ) then the lower bound 1.2) of the error term in 1.1) remains the same. On the other hand the upper bound 1.3) of the error term in Theorem 1.1 decreases by a factor k N+1. This improvement allows us to sieve for coprimality and thus to prove asymptotics for the number of primitive lattice points. 2. Generalisation of weak admissibility and statement of the results 2.1. Generalised weak admissibility. Let S = m β) where m = m 1... m n ) N n β = β 1... β n ) 0 ) n and n N = { }. We write x i for the elements in R mi and x = x 1... x n ) for the elements in R m1 R mn = R N where n N := m i. i=1 We will always assume that N > 1. We set n t := β i. i=1 We use to denote the Euclidean norm and we write n Nm β x) := x i βi for the multiplicative β-norm on R N induced by S. Let C R N be a coordinate-tuple subspace i.e. i=1 C = C I := {x R N ; x i = 0 for all i I)} where I {1... n}. We fix such a pair S C) and for Γ R N and ϱ > 0 we define the quantities νγ ϱ) := inf{nm β x) 1/t ; x Γ\C x < ϱ} Nm β Γ) := lim νγ ϱ). ϱ As usual we always interpret inf = and > x for all x R. The above quantities in the special case when C = {0} and m i = β i = 1 for all 1 i n) were introduced by Skriganov in [16 17]. By a lattice in R N we always mean a lattice of rank N. 4 In the sense of the Haar measure on SON R).

4 4 MARTIN WIDMER Definition 1. Let Λ be a lattice in R N. We say Λ is weakly admissible for S C) if νλ ϱ) > 0 for all ϱ > 0. We say Λ is admissible for S C) if Nm β Λ) > 0. Note that weak admissibility for a lattice in R N depends only on the choice of C and m whereas admissibility depends on C and S = m β). Also notice that a lattice Λ in R N is weakly admissible or admissible) in the sense of Skriganov [17] if and only if Λ is weakly admissible or admissible) for S C) with C = {0} and m i = β i = 1 for all 1 i n). Let us give some examples to illustrate that our notion of weak admissibility captures new interesting cases not covered by Skriganov s notion of weak admissibility. Example 1. Let Θ Mat r s R) be a matrix with r rows and s columns and consider 5 [ ] Ir Θ 2.1) Λ = Z r+s = {p + Θq q); p q) Z r Z s }. 0 I s We take n = 2 m 1 = r m 2 = s and C = {x 1 x 2 ); x 2 = 0}. Then the lattice Λ is weakly admissible for S C) for every choice of β) if p + Θq 0 for every q 0. If β = 1 β) then Λ is admissible for S C) if we have 2.2) p + Θq q β c Λ for every p q) with q 0 and some fixed c Λ > 0. The above lattice Λ naturally arises when considering Diophantine approximations for the matrix Θ cf. Corollary 2.2). Recall that the matrix Θ is called badly approximable if 2.2) holds true with β = s/r. W. M. Schmidt [14] has shown that the Hausdorff dimension of the set of badly approximable matrices is full i.e. rs. Another example comes from the Minkowski-embedding of an ideal in a number field. Example 2. Suppose K is a number field with r real and s pairs of complex conjugate embeddings. Let σ : K R r C s be the Minkowski-embedding and identify C in the usual way with R 2. Set n=r+s C = {0} m i = β i = 1 for 1 i r and m i = β i = 2 for r + 1 i r + s. Now let A K be a free Z-module of rank N = r + 2s. Then Λ = σa is admissible in S C). In particular this generalises the examples of Skriganov for totally real number fields to arbitrary number fields K. Unlike in Skriganov s setting we can also consider cartesian products of such modules A j by using the embedding σ : K p R pr C ps that sends a tuple α to σ 1 α)... σ r+s α)). Now m i is p if σ i is real and 2p otherwise while n and β i remain unchanged. Again we get that Λ = σa 1 A p ) is an admissible lattice in S C) Generalised aligned boxes. Now we introduce the sets in which we count the lattice points. Essentially these are the sets that are distorted only in the directions of the coordinate axes. Let S C) be given and recall that C = C I. For = 1... n ) 0 ) n we consider the β-weighted geometric mean n = and we assume throughout this note that 2.3) We set i=1 βi i ) 1/t i for all i / I). max := max i 1 i n min := min i. 1 i n 5 Despite the row notation we treat the vectors as column vectors.

5 WEAK ADMISSIBILITY PRIMITIVITY O-MINIMALITY AND DIOPHANTINE APPROXIMATION5 For κ > 0 and M N we introduce the family of sets F κm := {S R N ; AS) LipN M κ diamas)) A GL N R)}. Here GL N R) denotes the group of invertible N N-matrices with real entries diam ) denotes the diameter ) denotes the topological boundary and the notation Lip ) is explained in Definition 2 Section 3. It is an immediate consequence of [22 Theorem 2.6] that every bounded convex set in R N lies in F κm for κ = 8N 5/2 and M = 1. We will also show Proposition 8.1) that if Z R d+n is definable in an o-minimal structure explained in Section 8) and each fiber Z T = {x; T x) Z} R N is bounded then each fiber Z T lies in F κz M Z for certain constants κ Z and M Z depending only on Z but not on T. This result provides another rich source of interesting examples and might be of independent interest. For 1 i n let π i : R N R mi be the projection defined by π i x) = x i. Except in Theorem 2.2 we now no longer assume that Z is an aligned box; instead we impose the following less restrictive conditions. We fix values κ and M and we assume throughout this article that Z R N is such that for all 1 i n 1) Z F κm 2) π i Z ) B yi i ) for some y i R mi. Here B yi i ) denotes the closed Euclidean ball in R mi about y i of radius i. As is well known see e.g. [18]) Z ) LipN M L) implies that Z is measurable Main results. Let S C) be given. For Γ R N we introduce the quantities and λ 1 Γ) := inf{ x ; x Γ\{0}} µγ ϱ) := min{λ 1 Γ C) νγ ϱ)}. If µγ ϱ) = then we interpret 1/µΓ ϱ) as 0. Finally for a lattice Λ R N we introduce the error term E Λ Z ) := #Z Λ) VolZ det Λ. Our first result is a sharp upper bound for E Λ Z ). Theorem 2.1. Suppose Λ is a weakly admissible lattice for S C) and define c 1 := M1 + κ)n 2N ) N. Then we have E Λ Z ) c 1 inf 0<B max µλ B) + ) N 1 max. B By considering suitable uniformly scaled parallelepipeds it is clear that the error term cannot be improved in this generality. However the situation becomes much more interesting when we restrict the sets Z to aligned boxes. In this case Skriganov conjectured [16 Remark 1.1] that his error term [ ) Theorem 1.1] for admissible lattices in his sense) is sharp. Skriganov s conjecture would follow from the expected sharp lower bound for the extremal discrepancy of sequences in the unit cube in R N see [16 Remark 2.2]); however this is a major open problem in uniform distribution theory proved only for N = 2 by Schmidt [15]. Therefore the sharpness of Skriganov s error term for admissible lattices is known only for N = 2. Here we are able to show that for weakly admissible lattices in our sense) the error term in Theorem 2.1 is sharp for N = 2 and N = 3.

6 6 MARTIN WIDMER Theorem 2.2. Suppose 2 n 3 m i = β i = 1 1 i n) hence N = n) and C = {x; x n = 0}. Then there exists an absolute constant c abs > 0 a unimodular weakly admissible lattice Λ for S C) and a sequence of increasingly distorted i.e. / max tends to zero) aligned boxes Z = [ 1 1 ] [ n n ] satisfying 2.3) whose volume 2) N tends to infinity such that for each box Z E Λ Z ) c abs inf 0<B max µλ B) + ) N 1 max. B Thanks to the good dependence on the lattice of the error term in Theorem 2.1 we are also able to prove asymptotics for the number of primitive lattice points. Let Λ be a lattice in R N. We say x Λ is primitive if x is not of the form ky for some y Λ and some integer k > 1. We write Λ := {x Λ; x is primitive}. To state our next result let T : [0 ) [1 ) be monotonic increasing and an upper bound for the divisor function i.e. T k) d k 1 for all k N. Finally ζ ) denotes the Riemann zeta function. Theorem 2.3. Suppose Λ is a weakly admissible lattice for S C). Then there exists a constant c 2 = c 2 N κ M) such that #Z Λ ) VolZ ) N 1 ) ) ζn) det Λ c 2 µ µ + 1 T H) where 1 H = N 2N+2 + φy) ) µ + 1 ) µ = µλ max ) φy) is the Euclidean norm of y 1 / 1... y n / n ) R N and the vectors y i are those in 2) Subsection 2.2. Note that for every a > 2 there is a b = ba) expexp1)) such that for x b we can take T x) = a log x log log x. We use + φy) 1 + y /min ) and 1/µ + 1/ 2/µ to obtain the following corollary. Corollary 2.1. Suppose Λ is a weakly admissible lattice for S C) and a > 2. Then there exist constants c 3 = c 3 a N κ M y ) and b = ba) such that for all bµ we have #Z Λ ) VolZ ζn) det Λ c 3 µ where µ = µλ max ) and η = 1 + y / min. ) N 1 + a logη/µ) log logη/µ) ) ) µ Next we consider applications to Diophantine approximation. Let Θ Mat r s R) be a matrix with r rows and s columns and suppose that ϕ : [1 ) 0 1] is a nonincreasing function such that 2.4) p + Θq q β ϕ q )

7 WEAK ADMISSIBILITY PRIMITIVITY O-MINIMALITY AND DIOPHANTINE APPROXIMATION7 for every p q) with q 0. Let y be in R r 1 and let 0 < ɛ 1. We consider the system 2.5) 2.6) p + Θq y [0 ɛ] r q [0 ] s. Let NΘy ɛ ) be the number of p q) Zr+s that satisfy the above system and have coprime coordinates i.e. gcdp 1... p r q 1... q s ) = 1. In the one-dimensional case r = s = 1 Chalk and Erdős [4] proved in 1959 that if Θ is an irrational number and ɛ = ɛq) = 1/q)log q/ log log q) 2 then 2.5) has infinitely many coprime solutions i.e. NΘy ɛ ) is unbounded as tends to infinity. No improvements or generalisations have been obtained since. The following corollary follows straightforwardly from Corollary 2.1 and we leave the proof to the reader. We suppose ɛ = ɛ) is a function of and that ɛ β tends to infinity as tends to infinity. Corollary 2.2. Suppose a > 2. Then as tends to infinity we have where u = ɛ β ϕ) N Θyɛ ) = ) 1/1+β) and δ = ɛr s ζr + s) + Our+s 1 + ua log δ log log δ ) 1 ϕ) ɛ ) β ) 1/1+β). Corollary 2.2 also implies new results on how quickly ɛ can decay so that 2.5) still has infinitely many coprime solutions. As an example let us suppose that Θ is a badly approximable matrix so that in 2.4) we can choose β = s/r and ϕ ) to be constant. A straightforward computation shows that if c > 2 rs+s2 )/r 2 r+s 1)) and ɛ = ɛ) = s/r c log / log log then NΘy ɛ ) tends to infinity as does. In particular if ɛ = ɛ q ) = q s/r c log q / log log q then 2.5) has infinitely many coprime solutions 6. To the best of the author s knowledge this is the first such result in arbitrary dimensions. A similar simple calculation shows that Corollary 2.2 in conjunction with the classical Khintchine Groshev Theorem implies that the same holds true not only for badly approximable matrices Θ but for almost 7 every Θ Mat r s R). Finally we mention a connection to a question of Dani Laurent and Nogueira [5 10]. Suppose ɛ : [1 ) 0 1] and s 1 ɛ) r is non-increasing. Dani Laurent and Nogueira conjecture 8 [5 2. paragraph after Theorem 1.1] that if j N js 1 ɛj) r = then for almost every Θ Mat r s R) there exist infinitely many coprime solutions of 2.5) where again we interpret ɛ = ɛ q ) as a function evaluated at q. We cannot prove this conjecture but as mentioned before our result shows at least that we have infinitely many such solutions for almost every Θ if ɛ) s/r c log / log log and c > 2 rs+s2 )/r 2 r+s 1)). 3. Basic counting principle Let D 2 be an integer. Let Λ be a lattice of rank D in R D. Recall that B P R) denotes the closed Euclidean ball about P of radius R. We define the successive minima λ 1 Λ)... λ D Λ) of Λ as the successive minima in the sense of Minkowski with respect to the Euclidean unit ball. That is λ i = inf{λ; B 0 λ) Λ contains i linearly independent vectors}. 6 Here denotes the maximum norm. 7 With respect to the Lebesgue measure. 8 In fact their conjecture is more general but the mentioned special case is probably the most natural case.

8 8 MARTIN WIDMER Definition 2. Let M be a positive integer and let L be a non-negative real number. We say that a set S is in LipD M L) if S is a subset of R D and if there are M maps φ 1... φ M : [0 1] D 1 R D satisfying a Lipschitz condition φ i x) φ i y) L x y for x y [0 1] D 1 i = 1... M such that S is covered by the images of the maps φ i. For any set S we write 1 S) = { 1 if S 0 if S =. We will apply the following basic counting principle. Lemma 3.1. Let Λ be a lattice in R D with successive minima λ 1... λ D. Let S be a set in R D such that the boundary S of S is in LipD M L) and suppose S B P L) for some point P. Then S is measurable and moreover ) VolS D 1 #S Λ) L det Λ c 4D)M + 1 S Λ)) where c 4 D) = D 3D2 /2. Proof. By [21 Theorem 5.4] the set S is measurable and moreover { VolS #S Λ) det Λ /2 L j } 3.1) M max 1 D3D2. 1 j<d λ 1 λ j First suppose L λ 1. Then the lemma follows immediately from 3.1). Next we assume L < λ 1. We distinguish two subcases. First suppose S Λ. Then { L j } ) D 1 L max 1 = 1 = 1 S Λ) + 1 S Λ). 1 j<d λ 1 λ j λ 1 Now suppose S Λ =. As L < λ 1 we get using Minkowski s second Theorem VolS #S Λ) det Λ = VolS ) D 1 det Λ 2L)D L 2 D. λ 1 λ D This proves the lemma. Set λ 1 4. Proof of Theorem 2.1 Let θ i = / i 1 i n) and let φ be the automorphism of R N defined by Note that by 2.3) we have 4.1) Moreover and hence 4.2) φx) := θ 1 x 1... θ n x n ). θ min := min 1 i n θ i = / max. θ i 1 for all i / I). n i=1 θ βi i = 1 Nm β φx) = Nm β x). Lemma 4.1. We have φz ) LipN M L) for L = 2n 1/2 κ. λ 1

9 WEAK ADMISSIBILITY PRIMITIVITY O-MINIMALITY AND DIOPHANTINE APPROXIMATION9 Proof. We have φz ) φb y1 1 ) B yn n )) = B θ1y 1 ) B θny n ) and hence φz ) B φy n 1/2 ). As Z F κm the claim follows. Lemma 4.2. The set Z is measurable and #Z Λ) VolZ ) N 1 det Λ c φz λ 1 φλ) φλ)) where c 5 = 1 + 2n 1/2 κ) N 1 Mc 4 N). Proof. Since #Z Λ) = #φz φλ) and VolZ /det Λ = VolφZ /det φλ this follows immediately from Lemma 3.1 and Lemma 4.1. Lemma 4.3. Let B > 0. Then we have λ 1 φλ) min{λ 1 Λ C I ) νλ B) θ min B}. Proof. By 4.1) we have θ i 1 for all i / I). Moreover if x Λ C I then x i = 0 for all i I) and thus φx) 2 = θ i x i 2 x i 2 = x 2. 1 i n i / I 1 i n i / I Hence if x Λ C I and x 0 then φx) λ 1 Λ C I ). Now suppose that x Λ\C I. If z is an arbitrary point in R N then by the weighted arithmetic geometric mean inequality we have and thus 4.3) z 2 = n z i 2 1 max i β i i=1 n n ) 1 t β i z i 2 t z i 2βi max i β i i=1 Using 4.3) and 4.2) we conclude that z Nm β z) 1/t. i=1 φx) Nm β φx) 1/t = Nm β x) 1/t. First suppose that x < B. Then we have by the definition of ν ) Nm β x) 1/t νλ B) and hence φx) νλ B). Now suppose x B. Then we have Nm β z) 2/t φx) = θ min θ 1 x 1 /θ min... θ n x n /θ min ) θ min x 1... x n ) = θ min x θ min B. This proves the lemma. We can now easily finish the proof of Theorem 2.1. Since θ min max = we conclude λ 1 φλ) min{µλ B) B/ max }. Thus we have 4.4) λ 1 φλ) µλ B) + max B. The latter in conjunction with Lemma 4.2 and the fact c 5 +1 = 1+2n 1/2 κ) N 1 MN 3N 2 /2 + 1 M1 + κ)n 2N ) N = c 1 proves the theorem.

10 10 MARTIN WIDMER 5. Preparations for the Möbius inversion Recall that T : [0 ) [1 ) is a monotonic increasing function that is an upper bound for the divisor function i.e. T k) d k 1 for all k N. In this section D is a positive integer. For A GL D R) we write A for the Euclidean) operator norm. Lemma 5.1. Let Λ be a lattice in R D and let A be in GL D R) with AZ D = Λ. Then #{k N; B P R)\{0} kλ } T R + P ) A 1 )2R A 1 + 1). Proof. First assume A = I D so that Λ = Z D. Suppose v = a 1... a D ) Z D is nonzero kv B P R) and P = x 1... x D ). Then ka i lies in [x i R x i + R] for 1 i D. As v 0 there exists an i with a i 0. We conclude that k is a divisor of some nonzero integer in [x i R x i + R]. There are at most 2R + 1 integers in this interval each of which of modulus at most R + P. Hence the number of possibilities for k is T R + P )2R + 1). This proves the lemma for A = I D. Next note that #B P R)\{0} kλ) = #A 1 B P R)\{0} kz D ). Hence the general case follows from the case A = I D B A 1 P )R A 1 ) and A 1 P ) A 1 P. upon noticing A 1 B P R) Next we estimate the operator norm A 1 for a suitable choice of A. Lemma 5.2. Let Λ be a lattice in R D. There exists A GL D R) with AZ D = Λ and where c 6 D) = D 2D+1. A 1 c 6D) λ 1 Proof. Any lattice Λ in R D has a basis v 1... v D with v1 v D det[v 1...v D ] D2D see e.g. [21 Lemma 4.4]. Let A be the matrix that sends the canonical basis e 1... e D to v 1... v D. Now suppose A 1 sends e i to ϱ 1... ϱ D ) then by Cramer s rule ϱ j = det[v 1... e i... v D ] det[v 1... v j... v D ] det[v 1... e i... v D ] D 2D. v 1 v j v D Now we apply Hadamard s inequality to obtain det[v 1... e i... v D ] v 1 v j v D v 1 e j v D v 1 v i v D = 1 v i 1 λ 1. Next we use that for a D D matrix [a ij ] with real entries we have [a ij ] D max ij a ij and this proves the lemma. We combine the previous two lemmas. Lemma 5.3. Let Λ be a lattice in R D and let λ 1 = λ 1 Λ). Then 1 B P R)\{0} kλ) T k=1 c 6 D) R + P λ 1 )) ) 2c6 D)R + 1. λ 1 Proof. Note that k=1 1 B P R)\{0} kλ) = #{k N; B P R)\{0} kλ }. Hence the lemma follows immediately from Lemma 5.1 and Lemma 5.2.

11 WEAK ADMISSIBILITY PRIMITIVITY O-MINIMALITY AND DIOPHANTINE APPROXIMATION11 and Set Lemma 6.1. We have #Z Λ) VolZ det Λ c 7 where c 7 = 1 + 2n 1/2 κ) N 1 M + 1)c 4 N). 6. Proof of Theorem 2.3 Z := Z \{0} R := n 1/2. ) N B φy)r)\{0} φλ)) λ 1 φλ) Proof. Lemma 4.1 implies that Z LipN M + 1 L) with L = 2n1/2 κ. As noted in the proof of the latter lemma we have φz ) B φyr)\{0}. We conclude as in Lemma 4.2. For x Λ\{0} we define gcdx) := d if x = dx for some x Λ but x kx for all integers k > d and all x Λ. An equivalent definition is gcdaz) := gcdz) where z Z N gcdz) := gcdz 1... z N ) and Λ = AZ N.) Next we define F d) = {x Λ Z ; gcdx) = d}. In particular Λ Z = F 1). Then for k N we have the disjoint union F d) = kλ Z. k d If x = kx lies in kλ Z then kφx lies in kφλ B φy) R) and hence k R + φy) λ 1 φλ) R + φy) µλ max ) + R + φy) =: G where for the second inequality we have applied Lemma 4.3 with B = max. We use the Möbius function µ ) and the Möbius inversion formula to get #Λ Z ) = #F 1) = k=1 µk) d k d #F d) = [G] k=1 µk) d k d [G] #F d) = µk)#kλ Z ). For the rest of this section we will write g h to mean there exists a constant c = cn M κ) such that g ch. Applying Lemma 6.1 with Λ replaced by kλ yields #Z Λ ) VolZ ζn) det Λ [G] ) N 1 [G] + 1 B φy) R)\{0} kφλ) + VolZ kλ 1 φλ) k N det Λ. k=1 k>g k=1 First we note that k N k>g and moreover k max{g1} VolZ det Λ k=1 { } 1 N R k N max{g 1} 1 N max λ 1 φλ) 1 = VolφZ det φλ VolB 0R) det φλ RN λ 1 φλ) N.

12 12 MARTIN WIDMER Combining both with 4.4) yields k>g VolZ k N det Λ Next we note that by Lemma 5.3 [G] R λ 1 φλ) 1 B φy) R)\{0} kφλ) T k=1 Moreover and k=1 λ 1 φλ) µλ max ) + 1. c 6 N) R + φy) ) ) 2c6 N)R λ 1 φλ)) λ 1 φλ)) + 1. ) 2c6 N)R λ 1 φλ)) + 1 µλ max ) + 1 R + φy) λ 1 φλ)) R + φy) µλ max ) + R + φy) = G. Since c 6 N)G < H we conclude that [G] ) 1 B φy) R)\{0} kφλ) T H) µλ max ) + 1. Finally [G] ) N 1 kλ 1 φλ) µλ max ) + 1 k=1 where L = ) N 1 [G] ) N 1 k 1 N µλ max ) + 1 L k=1 { max{logg) 1} if N = 2 1 if N > 2. If N > 2 then L = 1 and we are done. So suppose N = 2. Hence c 6 N) = 32. By assumption T x) 1 so that L T c 6 N)G) for G exp1). Now suppose G > exp1). Since T is monotonic and 2 [log 2 [32G]] 32G we have T 32G) [log 2 [32G]] + 1 log 2 32G 1) log G. Thus L T c 6 N)G) T H). This finishes the proof. 7. Lower bounds for the error term The main goal of this section is to prove Theorem 2.2. Throughout this section we assume that m i = β i = 1 1 i n) so that N = n = t 2 and that Λ is a unimodular weakly admissible for S C) but not admissible for S C). To simplify the notation we write Nm ) := Nm β ) and ν ) := νλ ). Let k 1 be a constant and {x j } j=1 = {x j1... x jn )} j=1 be a sequence of pairwise distinct elements in Λ\C satisfying We define Nmx j ) kν x j ) n. N j := aν x j ) n Z j := N j B xj c j := λ n 1 Λ B xj ) where a > 0 is a constant which will be specified later B xj B xj := [ x j1 x j1 ] [ x jn x jn ] denotes the 0-centered box

13 WEAK ADMISSIBILITY PRIMITIVITY O-MINIMALITY AND DIOPHANTINE APPROXIMATION13 and λ i Λ B xj ) are the corresponding successive minima. For 1 i n we choose the values 9 i = N j x ji for the set Z j so that the condition 2) from Subsection 2.2 holds and moreover 7.1) We also assume that our sets Z j Lemma 7.1. We have #Z j Λ) VolZ j ak) 1 n 1 n N n j. satisfy the condition 2.3) i.e. i for all i / I). N j /c j n)) n 1 2 n akn n 1 j. Moreover N j tends to infinity and / max tends to zero. Proof. Let v 1... v n 1 be linearly independent lattice points in λ n 1 Λ B xj )B xj. Then the lattice points n 1 l=1 m lv l with N j /c j n) m l N j /c j n) are all distinct and lie all in Z j. Since 2[N j /c j n)] + 1 N j /c j n) and VolZ j = 2) n 2 n akn n 1 j the claimed inequality follows at once. Recall that Λ is not admissible. Therefore Nmx j ) and / max = Nmx j ) 1/n / max i { x ji } both tend to 0 and N j tends to infinity. We now make the crucial assumption that the n 1-th successive minimum c j is uniformly bounded 10 in j. Lemma 7.2. Suppose there exists a constant c Λ 1 such that 7.2) c j c Λ for all j and take a := 1/4k2c Λ n) n 1 ). Then we have 7.3) E Λ Z j ) #Z j Λ) VolZ j Proof. This follows immediately from Lemma ) c Λ n) n N n 1 j. Next we prove a general criterion for Λ under which we have #Z j Λ) VolZ j c inf 0<B max with a certain constant c > 0. Proposition 7.1. Suppose that the condition 7.2) and ) xj 7.5) ν ν x j ) n γν x j ) µλ B) + max B ) N 1 for some constant γ > 0 hold true. Then there exists c = ck c Λ n γ) > 0 such that 7.4) holds true for all j large enough. Proof. We have max N j x j and so ignoring the first few members of the sequence x j we can assume that Hence inf 0<B max µλ max ) νn j x j ) = νa x j /ν x j ) n ) ν x j /ν x j ) n ) γν x j ). µλ B) + ) max B µλ max ) + 1 ) ) γν x j ) + 1 2k 1/n /γ)n j for all j large enough. This in conjunction with 7.3) shows that 7.4) holds true. 9 To simplify the notation we suppress the dependence on j and we simply write i and. 10 Note that λ1 Λ B xj ) 1 by definition of the box B xj. On the other hand VolB xj tends to zero so that by Minkowski s second Theorem λ nλ B xj ) as j tends to infinity.

14 14 MARTIN WIDMER 7.6) For the rest of this section we assume that C = {x; x n = 0}. We now apply Proposition 7.1 to prove the case n = 2 in Theorem 2.2. Proposition 7.2. Suppose n = 2. Then there exists a unimodular weakly admissible lattice Λ for S C) and a sequence of increasingly distorted i.e. / max tends to zero) aligned boxes Z = [ 1 1 ] [ 2 2 ] whose volume 2) 2 tends to infinity such that E Λ Z ) c abs inf 0<B max µλ B) + ) max B where c abs > 0 is an absolute constant. Proof. Let α be an irrational real number and consider the lattice Λ given by the vectors p qα q) with p q Z. Then Λ is unimodular and weakly admissible for S C). To choose an appropriate α we consider its continued fraction expansion α = [a 0 a 1 a 2...]. Using the recurrence relation q j+1 = a j+1 q j + q j 1 for the denominator q j of the j-th convergent p j /q j in lowest terms) we can define α by setting a 0 = a 1 = 1 so that q 0 = q 1 = 1) and a j+1 = [log q j ] + 1. Next we note that a j+1 = [loga j q j 1 + q j 2 )] + 1 loga j +1)q j 1 )+1 loga j +1)+a j +1 3a j. Similarly we find a j +log a j 1 a j+1 and hence a j + log a j 1 a j+1 3a j. Put x j = p j q j α q j ) Λ\C so that x j > x j 1 at least for j large enough. From the theory of continued fractions we know that for x Λ\C the inequality Nmx) < 1/2 implies that x = cx j for some non-zero integer c and j N. We conclude that for all sufficiently large ϱ we have νϱ) 2 = Nmx j ) for some j. Also by the theory of continued fractions we know that 1/a j+1 + 2) < Nmx j ) < 1/a j+1. Since for j sufficiently large a j > a j we conclude Nmx j 1 ) < Nmx j 2 ) and thus Nmx j 1 ) = ν x j ) 2 for j large enough; so we can take k = 1. We also easily find that x j /ν x j ) 2 x j+1 for j large enough. It is now straightforward to verify 7.5). Moreover for j large enough 2.3) holds true and so Z j is an eligible set. Since n = 2 we automatically have 7.2) with c Λ = 1. Hence we can apply Proposition 7.1. Finally we note that VolZ j = 4Nj 2Nmx j ) = 2a)2 Nmx j 1 ) 2 Nmx j ) 2 6 a 2 j /a j+1 + 2) which tends to infinity and moreover that the boxes Z j are increasingly distorted by Lemma 7.1. This completes the proof. Next we prove the case n = 3 in Theorem 2.2. This case does not rely on Proposition 7.1. Proposition 7.3. Suppose n = 3. Then there exists a unimodular weakly admissible lattice Λ for S C) and a sequence of increasingly distorted aligned boxes Z = [ 1 1 ] [ 2 2 ] [ 3 3 ] whose volume 2) 3 tends to infinity such that E Λ Z ) c abs inf 0<B max µλ B) + ) 2 max B where c abs > 0 is an absolute constant.

15 WEAK ADMISSIBILITY PRIMITIVITY O-MINIMALITY AND DIOPHANTINE APPROXIMATION15 Proof. Let α = [a 0 a 1 a 2...] be a badly approximable real number so that the partial quotients a i are bounded. We set a M = max a i and we consider the lattice 7.7) Λ = {p 1 qα p 2 qα q); p 1 p 2 q Z}. The lattice Λ is unimodular and weakly admissible for S C). In this proof we write h g to mean h cg for a constant c = ca M ) depending only on a M. First we note that for every x Λ\C. Hence Nmx) x 1 7.8) νϱ) ϱ 1/3. Now suppose p j /q j is the j-th convergent of α and put x j = p j q j α p j q j α q j ) Λ\C. Then for j large enough 2.3) holds true and so Z j is an eligible set. Since Nmx j ) x j 1 we also conclude that there exists k = ka M ) 1 such that Nmx j ) kν x j ) 3. Since q j+1 = a j+1 q j + q j 1 we get q j+1 q j and as is wellknown p j+1 q j+1 α < p j q j α. Furthermore p j q j ) and p j+1 q j+1 ) are linearly independent and thus x j and x j+1 are linearly independent. Hence we conclude c j := λ 2 Λ B xj ) 1 and thus by Lemma 7.2 we get E Λ Z j ) Nj 2. Moreover for j sufficiently large we have 7.9) and thus 7.10) x j 1 < x j x j 1 ν x j ) Nmx j 1 ) 1/3 x j 1 1/3 x j 1/3. Combining 7.8) 7.9) and 7.10) implies that Therefore we have Thus Nj 2 max = N j q j Nj 2 have and thus for all j large enough inf 0<B max ϱ 1/3 νϱ) ϱ 1/3. N j ν x j ) 3 x j q j x j ν x j ) 3 N j. µλ B) + max B 2/3 and due to 7.1) Nj. Hence with B = N j we νb) max B ) 2 max B ) 2 N 2 j E Λ Z j ). Hence we have shown that 7.4) holds true. Finally we observe that VolZ j = 8Nj 3Nmx j) Nj 2 which due to Lemma 7.1 completes the proof.

16 16 MARTIN WIDMER 8. F κm - Families via o-minimality In this section let d 1 and D 2 both be integers. For Z R d+d and T R d we write Z T = {x R D ; T x) Z} and call this the fiber of Z above T. For the convenience of the reader we quickly recall the definition of an o-minimal structure following [11]. For more details we refer to [23 11] and [20]. Definition 3. A structure over R) is a sequence S = S n ) n N of families of subsets in R n such that for each n: 1) S n is a boolean algebra of subsets of R n under the usual set-theoretic operations). 2) S n contains every semi-algebraic subset of R n. 3) If A S n and B S m then A B S n+m. 4) If π : R n+m R n is the projection map onto the first n coordinates and A S n+m then πa) S n. An o-minimal structure over R) is a structure over R) that additionally satisfies: 5) The boundary of every set in S 1 is finite. The archetypical example of an o-minimal structure is the family of all semi-algebraic sets. Following the usual convention we say a set A is definable in S) if it lies in some S n. A map f : A B is called definable if its graph Γf) := {x fx)); x A} is a definable set. Proposition 8.1. Suppose Z R d+d is definable in an o-minimal structure over R and assume further that all fibers Z T are bounded sets. Then there exist constants κ Z and M Z depending only on Z but independent of T ) such that the fibers Z T lie in F κz M Z for all T R d. 8.1) Suppose the set Z is defined by the inequalities f 1 T x) 0... f k T x) 0 where the f i are certain real valued functions on R d+d. If all these functions f i are definable in a common o-minimal structure then Z is definable in an o-minimal structure. This happens for instance if the f i T x) = f i T 1... T d x 1... x D ) are restricted analytic functions 11 or polynomials in z 1... z d+d and each z i {T m expt m ) x l expx l ); 1 m d 1 l D}. For more details and examples we refer to [ ]. For the proof of Proposition 8.1 we shall need the following lemma. The author is grateful to Fabrizio Barroero for alerting him to Pila and Wilkie s Reparameterization Lemma for definable families and its relevance for the lemma. Lemma 8.1. Suppose Z R d+d is definable in an o-minimal structure over R and assume further that all fibers Z T are bounded sets. Then there exist constants κ Z and M Z depending only on Z such that the boundary Z T lies in LipD M Z κ Z diamz T )) for every T R d. Proof. First note that if #Z T 1 then Z T lies in LipD 1 0). Hence it suffices to prove the claim for those T with #Z T 2. By replacing Z with the definable set {T x) Z; x y Z T )x y)} we can assume that #Z T 2 for all T πz) where π is the projection onto the first d coordinates. We use the existence of definable Skolem functions. By [20 Ch.6 1.2) Proposition] there exists a definable map f : πz) R D whose graph Γf) Z. The proof of said 1.2) Proposition actually shows that there 11 By a restricted analytic function we mean a real valued function on R n which is zero outside of [ 1 1] n and is the restriction to [ 1 1] n of a function which is real analytic on an open neighbourhood of [ 1 1] n.

17 WEAK ADMISSIBILITY PRIMITIVITY O-MINIMALITY AND DIOPHANTINE APPROXIMATION17 is an algorithmic way to construct the Skolem function f. We will use the fact that this choice of f is determined by Z and π and hence can be seen as part of the data of Z. Now we consider the set Z = {T y); T x) Z y = x ft )}. This set is again definable and each non-empty fiber contains the origin i.e. 0 Z T for all T πz). Next we scale the fibers and translate by the point y 0 = 1/2)1... 1) R D to get a new definable set whose fibers all lie in 0 1) D. We put Z = {T z); T y) Z z = 3 diamz T )) 1 y y 0 } recall that diamz T ) = diamz T ) > 0 since Z T has at least two points). We note that the graph of the function T diamz T ) from πz) to R is given by {T t) πz) R; φt t) u R)φT u) u < t)} where φt t) stands for x y Z T ) x y t). This shows that the aforementioned map is definable and hence so is Z. Also we have Z T 0 1)D for all T. By [3 Lemma 3.15] the set Z = {T w); w Z T } is also definable. The fibers of a definable set are again definable cf. [3 Lemma 3.1]) and hence by [20 Ch ) Corollary] we have dim Z T ) D 1. From Pila and Wilkie s Reparameterization Lemma for definable families [ Corollary] we conclude 12 that Z T lies in LipD M Z κ Z ) for all T R d with certain constants κ Z and M Z. Rescaling and retranslating gives Z T LipD M Z κ Z diamz T )). Finally we note that Z depends only on Z and f which itself can be seen as part of the data of Z so that the constants κ Z and M Z may be chosen to depend only on Z. This completes the proof of the lemma. We can now prove Proposition 8.1. Consider the set Z := {ϕ T x); ϕ GL D R) x ϕz T )}. This set is definable in the given o-minimal structure and we have Z ϕt ) = ϕz T ). Applying Lemma 8.1 to the fibers Z ϕt ) we conclude that there exist constants κ Z and M Z such that ϕz T ) lies in LipD M Z κ Z diamϕz T ))) for all ϕ T ) GL D R) R d. Note that Z depends only on Z so that M Z κ Z are depending only on Z and this completes the proof of Proposition 8.1. Acknowledgements It is my pleasure to thank Fabrizio Barroero Michel Laurent Arnaldo Nogueira Damien Roy and Maxim Skriganov for helpful discussions. Thanks also to the anonymous referee for the careful and very helpful report. I completed this article during a visiting professorship at Graz University of Technology and I thank the Institute of Analysis and Number Theory for its hospitality. References 1. F. Barroero Counting algebraic integers of fixed degree and bounded height Monatsh. Math. 175 no ) Algebraic s-integers of fixed degree and bounded height Acta Arith. 167 no ) F. Barroero and M. Widmer Counting lattice points and o-minimal structures Int. Math. Res. Not ) J. H. H. Chalk and P. Erdős On the distribution of primitive lattice points in the plane Canad. Math. Bull ) S. G. Dani M. Laurent and A. Nogueira Multi-dimensional metric approximation by primitive points Math. Z. 279 no ) C. Frei D. Loughran and E. Sofos Rational points of bounded height on general conic bundle surfaces to appear in Newton Amer. J. Math 2016) 39 pages. 7. C. Frei and M. Pieropan O-minimality on twisted universal torsors and Manin s conjecture over number fields Ann. Sci. Éc. Norm. Supér. 49 4) 2016) Using that the partial derivatives are uniformly bounded we can extend the domain of the parametrisation to [0 1] D 1 without altering the Lipschitz constant.

18 18 MARTIN WIDMER 8. C. Frei and E. Sofos Counting rational points on smooth cubic surfaces Math. Res. Lett. 23 no ) Generalised divisor sums of binary forms over number fields to appear in J. Inst. Math. Jussieu 2016) 33 pages. 10. M. Laurent and A. Nogueira Inhomogeneous approximation with coprime integers and lattice orbits Acta Arith. 154 no ) J. Pila and A. Wilkie The rational points of a definable set Duke Math. J ) T. Scanlon A proof of the André-Oort conjecture using mathematical logic [after Pila Wilkie and Zannier] Astérisque Séminaire Bourbaki Exposé ). 13. O-minimality as an approach to the André-Oort conjecture submitted 2016) 53 pages. 14. W. M. Schmidt Badly approximable systems of linear forms J. Number Theory ) Irregularities of distribution. VII Acta Arith ) M. M. Skriganov Constructions of uniform distributions in terms of geometry of numbers Algebra Analiz. 6 No ) Ergodic theory on SLn) diophantine approximations and anomalies in the lattice point problem Invent. math ) P. G. Spain Lipschitz: a new version of an old principle Bull. London Math. Soc ) N. Technau and M. Widmer A note on Skriganov s counting theorem submitted 2016) 15 pages. 20. Lou van den Dries Tame topology and o-minimal structures London Mathematical Society Lecture Note Series vol. 248 Cambridge University Press Cambridge M. Widmer Counting primitive points of bounded height Trans. Amer. Math. Soc ) Lipschitz class narrow class and counting lattice points Proc. Amer. Math. Soc. 140 no ) A. J. Wilkie O-Minimal Structures Séminaire Bourbaki Exposé ). Department of Mathematics Royal Holloway University of London TW20 0EX Egham UK address: martin.widmer@rhul.ac.uk