Positive integers: counterexample to W.M. Schmidt's conjecture
aa r X i v : . [ m a t h . N T ] A ug Positive integers:counterexample to W.M. Schmidt’s conjecture by Nikolay G. Moshchevitin Abstract.
We show that there exist real numbers α , α linearly independent over Z together with 1 such that for ev-ery non-zero integer vector ( m , m ) with m > and m > one has || m α + m α || > − (max( m , m )) − σ with σ = 1 . + . Let || ξ || denotes the distance from real ξ to the nearest integer. Let φ = √ . In [1] W.M. Schmidtproved the following result.
Theorem A. (W.M. Schmidt)
Let real numbers α , α be linearly independent over Z togetherwith 1. Then there exists a sequence of integer two-dimensional vectors ( x ( i ) , x ( i )) such that x ( i ) , x ( i ) > ;2. || α x ( i ) + α x ( i ) || · (max { x ( i ) , x ( i ) } ) φ → as i → + ∞ .W.M. Schmidt posed a conjecture that the exponent φ here may be replaced by − ε witharbitrary positive ε (see [2]). In this paper we show this conjecture to be false.Let σ = 1 . + be the largest real root of the equation x − x − x + 1 = 0 . (1) Theorem 1.
There exist real numbers α , α such that they are linearly independent over Z together with 1 and for every integer vector ( m , m ) ∈ Z with m , m > and max( m , m ) > one has || m α + m α || > (max( m , m )) σ . We would like to formulate a related result from our paper [3]. For a real γ > we define afunction g ( γ ) = φ + 2 φ − φ γ − . One can see that g ( γ ) is a strictly decresaing function and g (2) = 2 , lim γ → + ∞ g ( γ ) = φ. Research is supported by the grant RFBR No. 09-01-00371-a Γ define C (Γ) = 2 Γ φ − φ φ γ − . In [3] the following statement was proved.
Theorem B.
Suppose that real numbers α , α satisfy the following Diophantine condition. Forsome Γ ∈ (0 , and γ > the inequality || α m + α m || > Γ(max {| m | , | m |} ) γ (2) holds for all integer vectors ( m , m ) ∈ Z \ { (0 , } . Then there exists an infinite sequence of integertwo-dimensional vectors ( x ( i ) , x ( i )) such that x ( i ) , x ( i ) > ;2. || α x ( i ) + α x ( i ) || · (max { x ( i ) , x ( i ) } ) g ( γ ) C (Γ) for all i .Of course constants , and in Theorem 1 and in the definition of C (Γ) may be reduced. We shall deal with the Euclidean norm for simplicity reason. So we use | · | for the Euclidean normof two- or three-dimensional vectors. By angle( u , v ) we denote the angle between vectors u , v .Define τ = 1 + σ σ = 1 . + . (3)Note that στ − > τ. (4)Put ω = τ + 1 . (5) Fundamental Lemma.
There exist real numbers α , α ∈ R linearly independent together with1 over Z and such that there exisis a sequence of integer vectors m = (1 , , − , m ν = ( m ,ν , m ,ν , m ,ν ) ∈ Z , ν = 1 , , , ... satisfying the following conditions (i) – (v) . (i) For any ν > the triple m ν − , m ν , m ν +1 consists of linearly independent vectors, and eachtwo-dimensional sublattice L ν = h m ν , m ν +1 i Z is complete, that is Z ∩ span L ν = L ν , ν = 0 , , , , .... (ii) Define ζ ν = m ,ν + m ,ν α + m ,ν α , M ν = | m ν | . For every ν > one has M ων +1 ζ ν M ων +1 . (6)2iii) M and for every ν > one has M ν M ν +1 (7) and H ν M ν +1 H ν , H ν = M στ − ν . (8)(iv) For every ν > one has m ,ν · m ,ν < ; moreover for the vectors e = (cid:18) (cid:19) , e = (cid:18) (cid:19) and m ν = ( m ,ν , m ,ν ) ∈ Z one has angle( m ν , ± e j ) > , j = 1 , . (9)(v) For every ν > for vectors m ν = ( m ,ν , m ,ν ) , m ν +1 = ( m ,ν +1 , m ,ν +1 ) one has angle( m ν , ± m ν +1 ) > . We give a sketched proof of Fundamental Lemma in Section 6. It use standard argument relatedto an inductive construction of special singular (in the sense of A. Khintchine) vectors. Inequality(4) is of major importance. Many different properties of singular vectors are discussed in our recentsurvey [4].For every ν we define two-dimensional lattice Λ ν = h m ν , m ν +1 i Z ⊂ Z . Let D ν be the fundamentalvolume of the lattice Λ ν . Obviously D ν M ν M ν +1 . (10)From the condition (v) one has D ν > M ν M ν +1 . (11)In the sequel we use the following notation. For an integer vector m = ( m , m , m ) ∈ Z wedefine ζ = ζ ( m ) = m + m α + m α , m = m ( m ) = ( m , m ) ∈ Z and M = M ( m ) = | m | . In Sections 3,4,5 below we suppose that α , α are the numbers from Fundamental Lemma. We prove a lemma concerning a lower bound for the value of | ζ ( m ) | in the case when the vector m ∈ Z is linearly independent of vectors m ν , m ν +1 .Consider the segment I ν = (cid:2) (4 M ν M ν +1 ) /σ , M τν +1 / (cid:3) (12)3inequalities (4) and (7) show that the left endpoitnt of the segment is less than the right endpointindeed). Lemma 1.
Suppose that a vector m ∈ Z is linearly independent of vectors m ν , m ν +1 and M ∈ I ν . (13) Then | ζ ( m ) | > M − σ . Proof.Consider the determinant ∆ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m m m m ,ν m ,ν m ,ν m ,ν +1 m ,ν +1 m ,ν +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ ( m ) m m ζ ν m ,ν m ,ν ζ ν +1 m ,ν +1 m ,ν +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We see from (6, 5) that | ∆ | | ζ ( m ) | M ν M ν +1 + 4 M M − τν +1 . From the inequality M M τν +1 / which follows from (13) we see that M M − τν +1 / . That is why | ζ ( m ) | M ν M ν +1 > / . Now we take into account the lower bound for M from (13) and the lemmafollows. (cid:3) m ν , m ν +1 Condition (i) means that each integer vector m ∈ Z which is linearly dependent together with m ν , m ν +1 can be written in a form m = λ m ν + µ m ν +1 with integer λ and µ . So if m ∈ Z is linearly dependent together with m ν , m ν +1 then for “cutten”vectors we have the equality m = λ m ν + µ m ν +1 (14)with integer λ and µ . Lemma 2 . Suppose that the vector m = ( m , m , m ) ∈ Z satisfy the condition m , m > Suppose that vectors m , m ν , m ν + are linearly dependent for some ν . Then | ζ ( m ) | > − M − σ . Proof.We can split two-dimensional lattice Λ ν into a countable union of one-dimensional lattices Λ ν,µ in the following way: Λ ν = G µ ∈ Z Λ ν,µ , Λ ν,µ = { z = ( z , z ) ∈ Λ ν : z = λ m ν + µ m ν +1 , λ ∈ Z } . By the condition (iv) there is no non-zero points ( z , z ) ∈ Λ ν, satisfying z · z > .Suppose that µ = 0 . As the fundamental volume of Λ ν is equal to D ν we see that the Euclideandistance between any two neighbouring lines aff Λ ν,µ and aff Λ ν,µ +1 is equal to D ν / q m ,ν + m ,ν .That is why the conditions ( m , m ) ∈ Λ ν,µ , m , m > (15)4mply max( m , m ) > | µ | D ν M ν > | µ | M ν +1 (16)(in the last inequality we use (11)).From the other hand conditions (15) together with (9) from (iv) lead to the inequality | λ | > | µ | M ν +1 M µ for the coefficient λ from (14). So (we apply lower bound from (ii) for ζ ν and upper bound from (ii)for ζ ν +1 ) we see that | ζ ( m ) | = | λζ ν + µζ ν +1 | > | µ | (cid:18) M ν +1 M ν ζ ν − ζ ν +1 (cid:19) > | µ | (cid:18) M ν M τν +1 − M τ +1 ν +2 (cid:19) . We apply lower bound (7) from (iii) to get | ζ ( m ) | > | µ | M ν M τν +1 . Now lower bound (8) from (iii) gives | ζ ( m ) | > − − στ − | µ | M τ + στ − ν +1 = 2 − − στ − | µ | M − σν +1 > − | µ | M − σν +1 (17)(here we use the definition of σ as a root of (1) and (3) to see that τ + στ − = σ ). Now we combine(16,17) and (8) from the condition (iii) to get | ζ ( m ) | > | µ | σ M σ > M σ . Lemma 2 is proved. (cid:3)
We take α , α from Fundamental Lemma. Consider an integer vector m = ( m , m , m ) with m , m > . We may suppose that | ζ ( m ) | = || m α + m α || . If for some ν vectors m , m ν , m ν +1 (18)are linearly dependent then application of Lemma 2 proves Theorem 1. So we may suppose that alltriples (18) consist of linearly independent vectors for every ν > . Now to prove Theorem 1 we mayuse Lemma 1. It is enough to show that [ ν > I ν ⊃ [2 , + ∞ ) (segments I ν are defined in (12)). But this follows from the condition M and the inequality (4 M ν M ν +1 ) /σ M τν / . The last inequality is a corollary of the right inequality from (8). (cid:3) Fundamental Lemma: sketch of a proof
Let m ∈ Z be an integer vector. The formulation of Fundamental Lemma deals with the values of M = M ( m ) = | m | . To describe the ideas of the proof it is much more convenient to consider theEuclidean norm M = | m | of the vector m itself than the Euclidean norm M = | m | of the “cutten”vector m ∈ Z . Of cource values of M and M are of the same order for all integer vectors m underconsideration. We may assume that M M M .Let S = { x = ( x , x , x ) ∈ R : | x | = 1 } be the unit sphere. We construct a sequence of nested closed sets B ν ⊂ S by induction. Theirunique common point x ∗ = ( x ∗ , x ∗ , x ∗ ) ∈ T ν B ν will define real numbers α = x ∗ /x ∗ , α = x ∗ /x ∗ which satisfy the conclusion of Fundamental Lemma.The base of inductive process is trivial.To proceed the inductive step we suppose that the following objects are alredy constructed:1) primitive integer vectors m j = ( m ,j , m ,j , m ,j ) , j ν with M j = | m j | ; we suppose thatthese vectors satisfy conditions (iv), (v);2) vectors ξ j = ( ξ ,j , ξ ,j , ξ ,j ) ∈ S such that ξ j ⊥ m j , ξ j ⊥ m j +1 , j ν − , andone-dimensional linear subspases Ξ j = span ξ j , j ν − ;3) two-dimensional linear subspaces ℓ j = (cid:8) ( x , x , x ) ∈ R : m ,j x + m ,j x + m ,j x = 0 (cid:9) , j ν − , and two-dimensional affine subspaces ℓ j = (cid:26) ( x , x , x ) ∈ R : m ,j x + m ,j x + m ,j x = 12M ωj +1 (cid:27) , j ν −
4) cylinders C j = (cid:26) x ∈ R : dist ( x , Ξ j ) ωj +1 M j (cid:27) , j ν − (here dist ( · , · ) denotes the Euclidean distance between sets) and closed sets G j = (cid:26) x ⊂ S ∩ C j : m ,j x + m ,j x + m ,j x > ωj +1 (cid:27) ⊂ S , j ν − (so a part of the boundary of G j belongs to ℓ j );5) two-dimensional complete sublattices L j = h m j , m j +1 i Z , j = 0 , ..., ν − with fundamentalvolumes d j satisfying inequalities M j M j +1 d j M j M j +1 (19)(here the right inequality is trivial, the left one means that the angle between vectors m j − , m j isbounded from below);6) we suppose that the vector m ν is defined, so we can consider linear subspace ℓ ν = (cid:8) ( x , x , x ) ∈ R : m ,ν x + m ,ν x + m ,ν x = 0 (cid:9) ; we suppose that linear subspases ℓ j for every j from the range j ν satisfy the condition ℓ j ∩ G j − = ∅ , j ν ; j from the range j ν there is a point η j = ( η ,j , η ,j .η ,j ) ∈ ℓ j ∩ G j − such that the set B j = (cid:26) x ∈ S : | x − η j | M ωj M j − (cid:27) satisfy the condition B j ⊂ G j − ⊂ B j − . (20)Here we should note that B ⊃ B ⊃ · · · ⊃ B ν − ⊃ B ν .We suppose that vectors m j , j ν and every couple α , α of the form α = x /x , α = x /x , x = ( x , x , x ) ∈ B ν − satisfy all the conditions (i) – (v) of Fundamental Lemma which aredefined up to the ( ν − -th step.Our task is to define an integer vector m ν +1 and all related objects of the ν -th step.Consider n = ( n , n , n ) ∈ Z such that the triple n , m ν − , m ν form a basis of Z . Such vectordoes exist as the lattice L ν − is complete. We may suppose that max( | n | , | n | ) M ν . (21)We consider two-dimensional lattices L ν − ,µ = { z = λ m ν − + λ m ν + µ n , λ , λ ∈ Z } . Note that Z = G µ ∈ Z L ν − ,µ . In fact n ∈ L ν − , . The Euclidean distance between the neighbouring affine subspaces aff L ν − ,µ and aff L ν − ,µ +1 is equal to d − ν − . Put µ ∗ = d ν − H ν M − ων M − ν − In fact µ is of the size µ ∗ ≍ M tν , t = στ − ω > (here the last inequality follows from (4)).Now here we define two-dimensional linear subspace ℓ ∗ ν ⊂ R and a point w ν ∈ aff L ν − ,µ ∗ by thefollowing way. Consider the unique one-dimensional affine subspace π ⊂ R such that1) η ν ∈ π ,2) π is parellel to ℓ ν − ,3) the intersection π ∩ S consists of just one point η ν .We define ℓ ∗ ν as follows: ℓ ∗ ν = span π. Let w ν ∈ aff L ν − ,µ ∗ be the unique point such that w ν ⊥ ℓ ∗ ν .Now we define the disk D ν = (cid:26) w ∈ aff L ν − ,µ ∗ : | w − w ν | H ν (cid:27) . Easy calculation shows that w ∈ D ν = ⇒ H ν | w | H ν . For w = ( w , w , w ) we consider two-dimensional linear subspase ℓ [ w ] = { x = ( x , x , x ) ∈ R : w x + w x + w x = 0 } . B ′ ν ⊂ B ν with the same center η ν and radius M − ων M ν − . Easy calculationshows that w ∈ D ν = ⇒ ℓ [ w ] ∩ ℓ ν ∩ B ′ ν = ∅ . Now if we take an integer vector m ν +1 ∈ D ν the conditions (ii), (iii) are satisfied for ν -th step.Vector ξ ν , subspaces ℓ ν , ℓ ν +1 and the set G ν are defined automatically. We can easily take η ν +1 with all necessary properties, in particular we condtruct B ν +1 (the second embedding in (20) with j = ν + 1 follows from the largeness of the value of M ν +1 , the first one can be ensured as the anglebetween vectors m ν +1 . m ν is almost the same as the angle between vectors m ν . m ν − ).Now we must explain how to ensure the condition (i). and (iv), (v).To get (i) we should note that a vector n = ( n , n , n ) ∈ L ν − , which completes the pair m ν − , m ν to a basis of Z may be found in any box of the form A k n k A k + M ν , k = 1 , (this fact follows from (21). For each n the vector m = µ ∗ n + m ν − ∈ L ν − ,µ ∗ together with m ν generates a complete lattice h m ν , m i Z . Note that M ν µ ∗ ≍ H ν · d ν − M τν M ν − = o ( H ν ) (we use the upper bound from (19)). So the set of all vectors m constructed is “dense” in the range H ν | m | H ν . So one can find such a vector with m ∈ D ν . That is why we can easy satisfy thecondition (i) for the ν -th step.Similarly, as we have many points m ∈ D ν satisfying (i) we can take m ν +1 close enough to w ν tosatisfy (iv), (v).As we have certain choice for the vector m ν at each step of the inductive construction we can get ( α , α ) satisfying linearly independence condition. So the point ( α , α ) constructed satisfies all theconditions of Fundamental Lemma. The inductive pocedure is described. References [1] W.M. Schmidt,
Two questions in Diophantine approximations , Monatshefte f¨ur Mathematik ,237 - 245 (1976).[2] W.M. Schmidt, Open problems in Diophantine approximations , in "Approximations Diophanti-ennes et nombres transcendants“ Luminy, 1982, Progress in Mathematics, Birkh¨auser, p.271 -289 (1983).[3] N.G. Moshchevitin,