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On Kleinbo k's Diophantine result 1 by Ni ky Mosh hevitin. Abstra t. We give an elementary proof of a re ent metri al Diophantine result by D. Kleinbo k related to badly approximable ve tors in a(cid:30)ne subspa es. Rd (x ,...,x ) 1 d R1.d+1De(cid:28)nitions and notation. Let be a(Exu ,lxid,e.a..n,xsp)a e wixth∈thRed oorxd∈inaRtde+s1 , 0 1 d let be a Eu lidean spa e with the oordinates . For or we denote |x| 9 by its sup-norm: 0 |x| = max |xj| |x| = max |xj|. 0 16j6d or 06j6d 2 A ∈ Rd A ∈ Rd+1 n Consider an a(cid:30)ne subspa e and de(cid:28)ne the a(cid:30)ne subspa e in the following way: u A = {x = (1,x ,...,x ) : (x ,...,x ) ∈ A}. J 1 d 1 d 8 B B ⊂ A ] Let be another a(cid:30)ne subspa e su h that . De(cid:28)ne T B = {x = (1,x ,...,x ) : (x ,...,x ) ∈ B}, B ⊂ A. N 1 d 1 d . h t Put a a = dimA = dimA, b = dimB = dimB. m [ We de(cid:28)ne linear subspa es A B = spanA, = spanB 1 v Rd+1 A B 1 as the smallest linear subspa es in ontaining or respe tively. So 4 A B 5 dim = a+1, dim = b+1. 1 . 6 ψ(T), T > 1 T → +∞ 0 Let be a positive valued fun tion de reasing to zero as . We de(cid:28)ne an B ψ 9 a(cid:30)ne subspa e to be -badly approximable if 0 : v 1 inf inf |x−y| > 0. Xi x∈Zd+1\{0} ψ(|x|) y∈B ! (1) r a This de(cid:28)nition is very onvenient for our exposition. B HerBe w=e{wwo}uld like to note that in the ase whenwt=he(aw(cid:30),n.e...s,uwbs)p∈a Red has zero dimension (and 1 d hen e onsists of just one nonzero ve tor ) the de(cid:28)nition (1) gives 1 inf inf |x−tw∗| > 0 x∈Zd+1\{0} ψ(|x|) t∈R (2) (cid:18) (cid:19) w∗ = (1,w ,...,w ) 1 d with . Inequality (2) is equivalent to the ondition that there exists a positive γ = γ(w) onstant su h that max ||w x|| > γ ·ψ(|x|), ∀x ∈ Z\{0} j 16j6d (3) || · || (here denotes the distan e to the nearest integer). The ondition (3) is a usual ondition of ψ -badly simultaneously approximable ve tor. This is an explanation our de(cid:28)nition (1). 1 supported by the grant RFBR (cid:157) 09-01-00371 1 ψ(T),ϕ(T) In the sequel we onsider two positive de reasing fun tions under the onditions ϕ(T) 6 ψ(T), lim ψ(T) = 0. T→+∞ R > 1 For these fun tions and de(cid:28)ne a−b a a T ϕ(RT) ϕ(R(T +1)) (R) (R) µ = , λ = − . T ψ(RT) T T T +1 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) 2. The result. Now we are ready to formulate the main result of this paper. R > 1 B ⊂ A 0 6 b = dimB < a = Theorem 1. Let . Suppose that are a(cid:30)ne subspa es and dimA 6 d B ψ ϕ(T) 6 ψ(T) . Let be a -badly approximable a(cid:30)ne subspa e. Suppose that and the series +∞ (R) (R) µ λ T T (4) T=1 X w ∈ A |w| 6 R onverges. Then almost all (in the sense of Lebesgue measure) ve tors su h that are ϕ -badly approximable ve tors. b = 0 ψ(T) = T−1/d ψ w We onsider a spe ial ase and . Then -badly approximable ve tors are known as simultaneously badly approximable ve tors (see [1℄, Chapter 2). Take ϕ(T) = ψ(T)·(logT)−∆. In the ase 1 ∆ > a the series (4) onverges. So we have the followAin⊂g Rd Corollary1. Considerana(cid:30)nesubspa e . Supposethat there existsabadly approximable w ∈ A ∆ > 1 A 1 ve tor . Suppose that a. Then almost all ve tors from the subspa e are T1/d(logT)∆-badly approximable ve tors. b = 1 ψ(T) = T−1/d Here we should note that in the ase and su h a result was obtained re ently by Dmitry Kleinbo k (see [2℄, Theorem 4.2) by means of theory of dynami s on homogeneous spa es. R > 1 3. Proof of Theorem 1. Given it is enough to prove that almost all ve tors w = (1,w ,...,w ) ∈ A |w | 6 R ϕ 1 d j su h that are -badly approximable in the sense of the de(cid:28)nition ≪,≍ (2). In the sequel we do not take are on the onstants. All onstants in the symbols may d A,B R depend on ,Csu⊂bsRpad+ e1s and x. ∈ Rd+1 |x|C x C For a set and a point we de(cid:28)ne the distan e from to by |x|C = inf |x−y|. C y∈ Consider the set Ω = {z ∈ Rd+1 : 0 6 z 6 T, max |z | 6 RT, |z|B 6 γ ·ψ(RT)} T 0 j 16j6d γ > 0 B ψ where is a lower bound for the in(cid:28)mum in (1). As is a -badly approximable subspa e we see that Ω ∩Zd+1 = {0} T 1/2 (we use the de(cid:28)nition (1)). Now we observe that any translation of the -dilated set 1 ·Ω +c, c ∈ Rd+1 T 2 (5) 2 x,y onsists of not more than one integer point. Indeed if two di(cid:27)erent integer points belong to the 0 6= x−y ∈ Ω T same set of the form (5) then . This is not possible. Consider the set Π = {z ∈ Rd+1 : 0 6 z 6 T, max |z | 6 RT, |z|A 6 ϕ(RT)}. T 0 j 16j6d ϕ(T) 6 ψ(T) ν ≪ µ(R) As we see that this set an be overed by not more than T T di(cid:27)erent sets of the Π T form (5). So we dedu e a on lusion about an upper bound for the number of integer points in : # Π ∩Zd+1 ≪ µ(R). T T (6) T > 1 (cid:0) (cid:1) For an integer we onsider the set Z = {z = (z ,z ,...,z ) ∈ Zd+1 : z = T, max |z | 6 RT, |z|A 6 ϕ(RT)} T 0 1 d 0 j 16j6d of the ardinality ζ(R) = #Z , 0 6 ζ(R) ≪ Ta. T T T (7) ρ > 0 z ∈ Rd+1 For and put U (z) = {y ∈ Rd+1 : |z−y| 6 ρ}. ρ Consider the set U U = (z) T ϕ(RT) z [ z where the union is taken over all integer points su h that z = T, max |z | 6 RT, U (z)∩A 6= ∅. 0 j ϕ(RT) 16j6d Clearly U U = (z) T ϕ(RT) z[∈ZT Consider the one G = {x ∈ Rd+1 : x = t·y, t ∈ R,y ∈ U } T T and the proje tion G U = A∩ . T T Consider the series +∞ mes U a T (8) T=1 X mes a a where stands for the -dimensionalLebesgue measure. By the Borel-Cantellilemma arguments Theorem 1 follows froUm the onvergen e of the series (8()R.) ζ ϕ(RT) Note that the set T is a union of not more than T balls (in sup-norm) of the radius . (R) U ⊂ A ζ ϕ(RT)/T Hen e the set T an be overed by not more than T balls of the radius . So in order to prove the onvergen e of the series (8) one an establish the onvergen e of the series +∞ a ϕ(RT) (R) ζ · . T T (9) T=1 (cid:18) (cid:19) X 3 It follows from (6) that T (R) (R) ζ ≪ µ . j T j=1 X Now the onvergen e of the series (9) follows from the onvergen e of the series (4) by partial summation as from (7) we see that a ϕ(R(T +1)) ζ(R) · ≪ (ϕ(R(T +1)))a → 0, T → +∞. T T +1 (cid:18) (cid:19) Theorem 1 is proved. Referen es [1℄ W.M. S hmidt, Diophantine Approximations, Le t. Not. Math., 785 (1980) [2℄ D. Kleinbo k, Extremal supspa es and their submanifolds, GAFA 13:2 (2003), 437 - 466. 4