Splitting transference inequalities with the help of Wolfgang Schmidt's parametric geometry of numbers
aa r X i v : . [ m a t h . N T ] M a y Splitting transference inequalitieswith the help ofWolfgang Schmidt’s parametric geometry of numbers. ∗ Oleg N. German
Abstract
This paper is a sequel to our previous paper arXiv:1105.1554, where we defined two typesof intermediate Diophantine exponents, connected them to Schmidt exponents and split Dyson’stransference inequality into a chain of inequalities for intermediate exponents. Here we presentsplitting of some other transference inequalities involving both regular and uniform Diophantineexponents.
Given a matrix Θ = θ · · · θ m ... . . . ... θ n · · · θ nm , θ ij ∈ R , n + m > , consider the system of linear equations Θ x = y (1)with variables x ∈ R m , y ∈ R n . The classical measure of how well the space of solutions to thissystem can be approximated by integer points is defined as follows. Let | · | denote the sup-norm inthe corresponding space. Definition 1.
The supremum of the real numbers γ , such that there are arbitrarily large values of t for which (resp. such that for every t large enough) the system of inequalities | x | t, | Θ x − y | t − γ (2)has a nonzero solution in ( x , y ) ∈ Z m ⊕ Z n , is called the regular (resp. uniform ) Diophantine exponent of Θ and is denoted by b (resp. a ).The transference principle connects the problem of approximating the space of solutions to (1) tothe analogous problem for the system Θ ⊺ y = x , (3)where Θ ⊺ denotes the transpose of Θ. Let us denote the Diophantine exponents corresponding to Θ ⊺ by b ∗ and a ∗ , respectively. ∗ This research was supported by RFBR (grant N ◦ ◦ MK–1226.2010.1. b in terms of b ∗ , and a in terms of a ∗ belongto A. Ya. Khintchine, V. Jarn´ık, F. Dyson and A. Apfelbeck. In the next Section we remind some ofthese results and formulate their improvements obtained recently by M. Laurent, Y. Bugeaud and alsoby the author. Then, in Section 3, we remind the definition of intermediate Diophantine exponentsfrom [1] and formulate the main results of this paper splitting the inequalities given in Section 2. Afterthat, in Section 4 we remind the definition of Schmidt’s exponents and their relation to Diophantineexponents. Finally, in Section 5 we prove our main results.Notice that in [1] we defined Diophantine exponents of two types. However, in this paper we shallconfine our considerations to the exponents of the second type (in terminology of [1]), not mentioningthe exponents of the first type at all (except in this very paragraph). It is interesting whether any“splitting” results can be obtained for the exponents of the first type. In [2] A. Ya. Khintchine proved for m = 1 his famous transference inequalities b ∗ > n b + n − , b > b ∗ ( n − b ∗ + n , (4)which were generalized later by F. Dyson [3], who proved that for arbitrary n , m b ∗ > n b + n − m − b + m . (5)While (4) cannot be improved (see [4], [5]) if only b and b ∗ are considered, stronger inequalities canbe obtained if a and a ∗ are also taken into account. The corresponding result for m = 1 belongs toM. Laurent and Y. Bugeaud (see [6], [7]). They proved that if the system (1) has no non-zero integersolutions, then ( a ∗ − b ∗ (( n − a ∗ + 1) b ∗ + ( n − a ∗ b (1 − a ) b ∗ − n + 2 − a n − . (6)The inequalities (6) were generalized to the case of arbitrary n , m by the author in [8], where it wasproved for arbitrary n , m that if the space of integer solutions of (1) is not a one-dimensional lattice,then along with (5) we have b ∗ > ( n − b ) − (1 − a )( m − b ) + (1 − a ) , (7) b ∗ > ( n − b − ) − ( a − − m − b − ) + ( a − − , (8)with (7) stronger than (8) if and only if a < V. Jarn´ık and A. Apfelbeck proved literal analogues of (4) and (5) for the uniform exponents, i.e. with b , b ∗ replaced by a , a ∗ , respectively (see [9], [10]). They also obtained some stronger inequalities ofa more cumbersome appearance. Among them, lonely in its elegance, stands the equality a − + a ∗ = 1 (9)2roved by Jarn´ık for n = 1, m = 2. The results of Jarn´ık and Apfelbeck were improved by the authorin [8], where it was proved that for arbitrary n , m we have a ∗ > n − m − a , if a ,n − a − m − , if a > . (10) Set d = m + n . Denote by ℓℓℓ , . . . , ℓℓℓ m the columns of the matrix (cid:18) E m Θ (cid:19) , where E m is the m × m unity matrix. Clearly, ℓℓℓ , . . . , ℓℓℓ m span the space of solutions to the system(1). Let us set for each k -tuple σ = { i , . . . , i k } , 1 i < . . . < i k m , L σ = ℓℓℓ i ∧ . . . ∧ ℓℓℓ i k , (11)denote by J k the set of all the k -element subsets of { , . . . , m } , k = 0 , . . . , m , and set L ∅ = 1.Let us also set k = max(0 , m − p ). Definition 2.
The supremum of the real numbers γ , such that there are arbitrarily large values of t for which (resp. such that for every t large enough) the system of inequalitiesmax σ ∈J k | L σ ∧ Z | t − ( k − k )(1+ γ ) , k = 0 , . . . , m, (12)has a nonzero solution in Z ∈ ∧ p ( Z d ) is called the p -th regular (resp. uniform) Diophantine exponent of Θ and is denoted by b p (resp. a p ).For m = 1 the quantities b p , a p were defined by Laurent in [11]. The consistency of Definitions 1and 2 for arbitrary n , m was proved in [1] (see Propositions 4, 5 therein). Laurent and Bugeaud usedthe exponents b p to split (4) into a chain of inequalities relating b p to b p +1 . Namely, they proved thatfor m = 1 we have b ∗ = b n and b p +1 > ( n − p + 1) b p + 1 n − p , b p > p b p +1 b p +1 + p + 1 , p = 1 , . . . , n − . (13)Besides that, they proved for m = 1 that if the system (1) has no non-zero integer solutions, then wehave a ∗ = a n and b > b + a − a , b n − > − a − n b − n + a − n , (14)which, combined with (13), gave them (6).In [1] we generalized (13) and its analogue for the uniform exponents to the case of arbitrary n , m . We showed that b ∗ p = b d − p , a ∗ p = a d − p , p = 1 , . . . , d − , (15)where b ∗ p and a ∗ p are p -th regular and uniform Diophantine exponents of Θ ⊺ , and proved3 heorem 1. For each p = 1 , . . . , d − the following statements hold.If p > m , then ( d − p − b p +1 ) > ( d − p )(1 + b p ) , (16)( d − p − a p +1 ) > ( d − p )(1 + a p ) . (17) If p m − , then ( d − p − b p ) − > ( d − p )(1 + b p +1 ) − − n, (18)( d − p − a p ) − > ( d − p )(1 + a p +1 ) − − n. (19)The first result of the current paper generalizes (14). We prove Theorem 2.
Suppose that the space of integer solutions of (1) is not a one-dimensional lattice. Thenfor m = 1 we have b > b + a − a , (20) and for m > we have b > a −
12 + b − a , if a = ∞ , − a − b − + a − . (21)The first inequality of (20) is exactly the first inequality of (14). The second inequality of (21) inview of (15) gives the second inequality of (14).It follows from Theorem 1 that for m > d − b d − ) − (1 + b ) − + m − . (22)Combining this inequality with (21) we get (7) and (8), in case m > Theorem 3.
For m = 1 we have a > (1 − a ) − − n − n − . (23) For m > we have a > n − − n − ( d − − a ) − , if a ,m − n + ( d − a − − , if a > . (24)Let us show that Theorem 3 splits (10) the very same way Theorem 2 splits (7) and (8). It followsfrom Theorem 1 that for m = 1 1 + a n > ( n − a ) (25)and that for m > d − a d − ) − (1 + a ) − + m − . (26)Combining (26) with (24), we get (10) for m >
2. As for m = 1, we always have a m = 1. 4 Schmidt’s exponents
We start this Section with reminding the definition of Schmidt’s exponents of the second type we gavein [1] basing on [12].Let Λ be a unimodular d -dimensional lattice in R d . Denote by B d ∞ the unit ball in sup-norm, i.e.the cube with vertices at the points ( ± , . . . , ± d -tuple τττ = ( τ , . . . , τ d ) ∈ R d denote by D τττ the diagonal d × d matrix with e τ , . . . , e τ d on the main diagonal. Let us also denote by λ p ( M ) the p -th successive minimum of a compact symmetric convex body M ⊂ R d (centered at the origin) withrespect to the lattice Λ.Suppose we have a path T in R d defined as τττ = τττ ( s ), s ∈ R + , such that τ ( s ) + . . . + τ d ( s ) = 0 , for all s. (27)In our further applications to Diophantine approximation we shall confine ourselves to a path that isa ray with the endpoint at the origin and all the functions τ ( s ) , . . . , τ d ( s ) being linear.Set B ( s ) = D τττ ( s ) B d ∞ . Consider the functions ψ p (Λ , T , s ) = ln( λ p ( B ( s ))) s , p = 1 , . . . , d. Definition 3.
We call the quantitiesΨ p (Λ , T ) = lim inf s → + ∞ (cid:18) p X i =1 ψ i (Λ , T , s ) (cid:19) , Ψ p (Λ , T ) = lim sup s → + ∞ (cid:18) p X i =1 ψ i (Λ , T , s ) (cid:19) the p -th lower and upper Schmidt’s exponents , respectively.It appears that interpreting Diophantine exponents in terms of Schmidt’s exponents simplifiesmany constructions and reveals the nature of some phenomena. In order to deliver this interpretationlet us consider the latticeΛ = n(cid:16) h e , z i , . . . , h e m , z i , h ℓℓℓ m +1 , z i , . . . , h ℓℓℓ d , z i (cid:17) ⊺ ∈ R d (cid:12)(cid:12)(cid:12) z ∈ Z d o , (28)where e , . . . , e m are the first m columns of the d × d unity matrix, and ℓℓℓ m +1 , . . . , ℓℓℓ d are the columnsof the matrix (cid:18) − Θ ⊺ E n (cid:19) . Let us also consider the path T : s τττ ( s ) defined by τ ( s ) = . . . = τ m ( s ) = s, τ m +1 ( s ) = . . . = τ d ( s ) = − ms/n. (29)Thus, we have connected to Θ the exponents Ψ p (Λ , T ), Ψ p (Λ , T ), which we shall simply denote byΨ p and Ψ p . We can do the same thing to Θ ⊺ , and obtain the exponents we choose to denote by Ψ ∗ p and Ψ ∗ p .In [1] we proved the following statements: Proposition 1.
Set κ p = min( p, mn ( d − p )) . Then (1 + b p )( κ p + Ψ p ) = (1 + a p )( κ p + Ψ p ) = d/n. (30)5 roposition 2. Set κ ∗ p = min( p, nm ( d − p )) . Then (1 + b ∗ p )( κ ∗ p + Ψ ∗ p ) = (1 + a ∗ p )( κ ∗ p + Ψ ∗ p ) = d/m. (31)Notice that in view of (15) it follows from (30), (31) thatΨ ∗ p = nm Ψ d − p and Ψ ∗ p = nm Ψ d − p . (32)It was also shown implicitly in [1] that b p > a p > dn κ p − , which is equivalent to − κ p Ψ p Ψ p . Let us now translate Theorems 2, 3 into the language of Schmidt’s exponents. Theorem 2 turnsinto
Theorem 4.
Suppose that the space of integer solutions of (1) is not a one-dimensional lattice. Then Ψ + d · Ψ − Ψ n + n Ψ , if Ψ = − , + d · Ψ − Ψ m − n Ψ . (33)Theorem 3 turns into Theorem 5.
We have Ψ ( d − ( n −
1) + n Ψ , if Ψ > m − n n , ( d − ( m − − n Ψ , if Ψ m − n n . (34)As we see, this point of view relieves us of singling out the case m = 1. In the next Section weprove Theorems 4, 5. Having Λ and T fixed by (28), (29), let us write ψ p ( s ) instead of ψ p (Λ , T , s ). Let us also setΨ p ( s ) = p X i =1 ψ i ( s ) . In [1] we showed that 0 − Ψ d ( s ) = O ( s − ) , (35)whence we derived that for every p within the range 1 p d − p +1 d − p − Ψ p d − p and Ψ p +1 d − p − Ψ p d − p , (36)which is the very Theorem 1 reformulated in terms of Schmidt’s exponents. Now we shall need a moreprecise version of (36). 6 roposition 3. For every p within the range p d − and every s > we have p + 1 p Ψ p ( s ) Ψ p +1 ( s ) d − p − d − p Ψ p ( s ) . (37) Proof.
In view of (35), it follows from the inequalities ψ i ( s ) ψ i +1 ( s ), i = 1 , . . . , d −
1, that1 p p X i =1 ψ i ( s ) ψ p +1 ( s ) − d − p p X i =1 ψ i ( s ) , which immediately implies (37).The following observation is the crucial point for proving Theorems 4, 5. Lemma 1.
Suppose s, s ′ ∈ R + satisfy the conditions λ ( B ( s )) B ( s ) ⊆ λ ( B ( s ′ )) B ( s ′ ) , (38) λ ( B ( s ′ )) = λ ( B ( s ′ )) . (39) Then ψ ( s ) ψ ( s ) + d · ψ ( s ′ ) − ψ ( s ) n + nψ ( s ′ ) , if s ′ s and ψ ( s ′ ) = − ,ψ ( s ) + d · ψ ( s ′ ) − ψ ( s ) m − nψ ( s ′ ) , if s ′ > s. (40) Proof.
Suppose that s ′ s . Then it follows from (38) and (39) that λ ( B ( s )) e s = λ ( B ( s ′ )) e s ′ > λ ( B ( s )) e − ms/n λ ( B ( s ′ )) e − ms ′ /n = λ ( B ( s ′ )) e − ms ′ /n , i.e. s (1 + ψ ( s )) = s ′ (1 + ψ ( s ′ )) > s ( ψ ( s ) − m/n ) s ′ ( ψ ( s ′ ) − m/n ) (42)Combining (41) and (42) we get the first inequality of (40).Suppose now that s ′ > s . Then it follows from (38) and (39) that λ ( B ( s )) e − ms/n = λ ( B ( s ′ )) e − ms ′ /n < λ ( B ( s )) e s λ ( B ( s ′ )) e s ′ = λ ( B ( s ′ )) e s ′ , i.e. s ( ψ ( s ) − m/n ) = s ′ ( ψ ( s ′ ) − m/n ) < s (1 + ψ ( s )) s ′ (1 + ψ ( s ′ )) (44)Combining (43) and (44) we get the second inequality of (40).7or each z = ( z , . . . , z d ) ⊺ ∈ R d and each s > µ s ( z ) = e − s max i m | z i | and ν s ( z ) = e ms/n max m
For each s > , such that µ s ( v s ) = ν s ( v s ) = λ ( B ( s )) , (45) there are s ′ , s ′′ > , such that s (1 + ψ ( s )) s ′ s s ′′ s (1 − ( n/m ) ψ ( s )) and Ψ ( s ) ψ ( s ) + d · ψ ( s ′ ) − ψ ( s ) n + nψ ( s ′ ) , if ψ ( s ′ ) = − , ψ ( s ) + d · ψ ( s ′′ ) − ψ ( s ) m − nψ ( s ′′ ) . (46) Proof.
Let us show that the relation µ s ( v s ) = λ ( B ( s )) implies the existence of an s ′ s satisfyingthe conditions of Lemma 1. Denote λ = λ ( B ( s )). Let P ν = n z ∈ R d (cid:12)(cid:12)(cid:12) µ s ( z ) λ, ν s ( z ) νλ o be the minimal (w.r.t inclusion) parallelepiped containing no non-zero points of Λ in its interior. Theexistence of such a parallelepiped follows from Minkowski’s convex body theorem. It also implies that1 ν λ − d/n . Then λ B ( s ) ⊆ P ν = λ ′ B ( s ′ ) , where λ ′ = λν n/d , s ′ = s − ( n/d ) ln ν . For λ ′ , s ′ we have λ ′ > λ, s + ln λ s ′ s. On the other hand, P ν contains non-collinear points of Λ in its boundary, so λ ( B ( s ′ )) = λ ( B ( s ′ )) = λ ′ .Thus, s , s ′ satisfy (38), (39).Now let us consider the relation ν s ( v s ) = λ ( B ( s )). By Minkowski’s convex body theorem there isa µ in the interval 1 µ λ − d/m , such that the parallelepiped Q µ = n z ∈ R d (cid:12)(cid:12)(cid:12) µ s ( z ) µλ, ν s ( z ) λ o contains no non-zero points of Λ in its interior, but contains non-collinear points of Λ in its boundary.Then λ B ( s ) ⊆ Q µ = λ ′′ B ( s ′′ ) , where λ ′′ = λµ m/d , s ′′ = s + ( n/d ) ln µ . For λ ′′ , s ′′ we have λ ′′ > λ, s s ′′ s − ( n/m ) ln λ. Besides that, s , s ′′ also satisfy (38), (39), since λ ( B ( s ′′ )) = λ ( B ( s ′′ )) = λ ′′ .It remains to apply Lemma 1. 8aving Corollary 1, it is easy now to prove Theorem 4.First, let us notice that if the system (1) has a non-zero integer solution, then it has two linearlyindependent integer solutions, so in this case Ψ = Ψ = −
1, Ψ = −
2, which implies (33).Next, let us suppose that the system (1) has no non-zero integer solutions. Then there are infinitelymany local minima of ψ ( s ), each of them satisfies (45), and the sequence of these local minima tendsto ∞ . Moreover, s ′ and s ′′ from Corollary 1 tend to ∞ as s tends to ∞ . Indeed, since (1) has nonon-zero integer solutions, we have e s (1+ ψ ( s )) = e s λ ( B ( s )) = λ ( e − s B ( s )) → ∞ as s → ∞ , so s (1 + ψ ( s )) → ∞ as s → ∞ . (47)Particularly, it follows from (47) that ψ ( s ) is eventually greater than − ψ ( s ) > − ψ ( s )). Therefore,Ψ lim inf Ψ ( s ) ψ ( s ) + d · lim sup ψ ( s ′ ) − ψ ( s ) n + nψ ( s ′ ) , ψ ( s ) + d · lim sup ψ ( s ′′ ) − ψ ( s ) m − nψ ( s ′′ ) , (48)where the lim inf and the lim sup are taken over the set of local minima of ψ ( s ). Since ψ ( s ) is neverpositive, both denominators in (48) are eventually positive. Therefore, (48) implies (33). Corollary 2.
Suppose that the system (1) has no non-zero integer solutions. Then for each s > there is an s ′ > , such that s (1 + ψ ( s )) s ′ s (1 − ( n/m ) ψ ( s )) , and Ψ ( s ) ( d − ψ ( s ′ )( n −
1) + nψ ( s ′ ) , if ψ ( s ′ ) > m − n n , ( d − ψ ( s ′ )( m − − nψ ( s ′ ) , if ψ ( s ′ ) m − n n . (49) Proof.
Assume that µ s ( v s ) = λ ( B ( s )). Then the same argument as in the proof of Corollary 1 showsthat there is an s ′ , such that s (1 + ψ ( s )) s ′ s , andΨ ( s ) ψ ( s ) + d · ψ ( s ′ ) − ψ ( s ) n + nψ ( s ′ ) , (50)unless ψ ( s ′ ) = −
1. By Proposition 3 we have d − d − ( s ) ψ ( s )
12 Ψ ( s ) . (51)If ψ ( s ′ ) = −
1, then (51) implies (49). Suppose that ψ ( s ′ ) = −
1. Then, taking into account that2 − dn + nψ ( s ′ ) > ψ ( s ′ ) > m − n n , we conclude from (50) and (51) thatΨ ( s ) ψ ( s ′ ) , if ψ ( s ′ ) > m − n n , ( d − ψ ( s ′ )( m − − nψ ( s ′ ) , if ψ ( s ′ ) m − n n . (52)9ssume now that ν s ( v s ) = λ ( B ( s )). Then the same argument as in the proof of Corollary 1 showsthat there is an s ′′ , such that s s ′′ s (1 − ( n/m ) ψ ( s )), andΨ ( s ) ψ ( s ) + d · ψ ( s ′′ ) − ψ ( s ) m − nψ ( s ′′ ) . (53)Taking into account that2 − dm − nψ ( s ′′ ) > ψ ( s ′′ ) m − n n , we conclude from (53) and (51) thatΨ ( s ) ( d − ψ ( s ′′ )( n −
1) + nψ ( s ′′ ) , if ψ ( s ′′ ) > m − n n , ψ ( s ′′ ) , if ψ ( s ′′ ) m − n n . (54)Since ψ ( s ′ ) and ψ ( s ′′ ) are negative, we have2 ψ ( s ′ ) ( d − ψ ( s ′ )( n −
1) + nψ ( s ′ ) , if ψ ( s ′ ) > m − n n , and 2 ψ ( s ′′ ) ( d − ψ ( s ′′ )( m − − nψ ( s ′′ ) , if ψ ( s ′′ ) m − n n . Therefore, (52) and (54) imply the desired statement.Deriving Theorem 5 from Corollary 2 is even easier than deriving Theorem 4 from Corollary 1.If the system (1) has a non-zero integer solution, then Ψ = − < m − n n , and (34) follows from(36). Suppose now that (1) has no non-zero integer solutions. Then it follows from (47) that s ′ fromCorollary 2 tends to ∞ as s tends to ∞ . Hence, taking lim sup of both sides in (49), we get (34). References [1]
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German
Moscow Lomonosov State UniversityVorobiovy Gory, GSP–1119991 Moscow, RUSSIA
E-mail : [email protected], [email protected]@mech.math.msu.su, [email protected]