Restricted simultaneous Diophantine approximation
aa r X i v : . [ m a t h . N T ] J un RESTRICTED SIMULTANEOUS DIOPHANTINE APPROXIMATION
STEPHAN BAIER AND ANISH GHOSH
Abstract.
We study the problem of Diophantine approximation on lines in R d undercertain primality restrictions. Contents
1. Introduction 12. A Metrical approach 33. Proof of Theorem 2.2(i) 44. Proof of Theorem 2.2(ii) 13References 171.
Introduction
The subject of metric Diophantine approximation on manifolds studies Diophantine approxi-mation of typical points on submanifolds in R d by rational points in R d . This subject has receivedconsiderable attention in the last two decades, leading to dramatic progress using methods arisingfrom the ergodic theory of flows on homogeneous spaces as well as analytic methods. If one putsfurther restrictions on the approximating rationals, then the situation is much less understood.A very natural class of problems arises by imposing primality restrictions on the approximatingrationals. In this paper, we study the problem of Diophantine approximation for vectors onlines in R d with additional primality restrictions. Thus we combine the themes of simultaneousmetric Diophantine approximation on affine subspaces , and Diophantine approximation with re-strictions. Both problems have their own substantial complications and have separately receivedconsiderable attention, cf. [1, 6, 16] for Diophantine approximation on affine subspaces, and[10, 11, 12, 14, 15] for Diophantine approximation with primality constraints. The only previousworks on the combined theme that we are aware of are the work of Harman-Jones [13] regardingDiophantine approximation on curves in R with constraints, and our previous work [2] wherewe addressed the problem of Diophantine approximation with constraints on lines in R . Themain result of the present paper generalises [2] to arbitrary dimensions and is the first such resultin this generality. While the broad strategy in the present paper is similar to that of [2], thegreater generality makes the problem significantly more complicated. In studying Diophantine Mathematics Subject Classification. approximation on lines, and more generally, affine subspaces, it is natural and indeed imperativeto impose some Diophantine condition on the line or subspace itself. In [2], we had assumed thatthe slope of the line in R is irrational. In higher dimensions, the lack of a suitable continued frac-tion algorithm makes it necessary to replace irrationality with a suitable Diophantine conditionwhich we now introduce. Let k k denote the distance to the nearest integer of a real number, k k ∞ denote the supremum norm of a vector and for vectors v , c ∈ R d , denote by v · c = v c + ... + v d c d the inner product of v and c . Recall that c ∈ R d is called k -Diophantine ( c ∈ D k ( R d )) if thereexists a constant C > || v · c || > C || v || k ∞ for every v ∈ Z d \ { } . (1)Our main result is: Theorem 1.1.
Let d be a positive integer and k ≥ d be a positive real number. Define γ d,k := 1 d (3 k + 2) (2) and suppose that < ε < γ d,k . Let c , ..., c d be positive irrational numbers such that the vector c = ( c , ..., c d ) is k -Diophantine. Then for almost all positive real α , with respect to the Lebesguemeasure, there are infinitely many ( d + 2) -tuples ( p, q , ..., q d , r ) with p and r prime and q , ..., q d positive integers such that simultaneously < pα − r < p − γ d,k + ε , < pc i α − q i < p − γ d,k + ε for all i ∈ { , ..., d } . (3) Remarks :(1) It is well known that D d ( R d ) is a nonempty set of zero Lebesgue measure and full Haus-dorff dimension. These comprise the set of badly approximable vectors. Moreover, D k ( R d )has full measure whenever k > d , see [4] for example.(2) In [2], we proved the analogue of Theorem 1.1 for lines in R under the assumption thatthe slope c of the line is irrational. In fact, what was used was the following Diophantineproperty of irrational numbers: There exists an infinite set S of integers and a positiveconstant D > v ∈ Z d \{ }|| v || ∞ ≤ N || v · c || ≥ N − D if N ∈ S . (4)For d = 1 this holds for D = 1 and S being the set of numbers [ N/ N ’sare the denominators in the continued fraction approximants of c = c . This can be seenby approximating c by its continued fraction approximant a/N and using the fact that | c − a/N | ≤ /N . In the case d = 1, a sequence S with the above property can alsobe constructed using the Dirichlet approximation theorem and the condition that c isirrational. There is a d -dimensional version of Dirichlet’s approximation theorem, butfor d ≥
2, a statement like (4) does not follow from it. It is likely that the conclusionof Theorem 1.1 would also hold in arbitrary dimension by assuming (4) rather than theDiophantine condition we have assumed, with perhaps a different exponent. We notehowever, that for d = 1, choosing k = 1, we recover the exponent in [2]. The condition(4) is also an interesting Diophantine property, and can be shown to include the class of ESTRICTED SIMULTANEOUS DIOPHANTINE APPROXIMATION 3 nonsingular vectors.(3) It is an interesting problem to consider analogues of Theorem 1.1 for affine subspaces oflower codimension, and indeed for manifolds not contained in affine subspaces, the nondegenerate manifolds. We will consider these in a forthcoming work.
Acknowledgements.
S. Baier wishes to thank the Tata Institute of Fundamental Researchin Mumbai (India) for its warm hospitality and excellent working conditions. This work wascompleted while Ghosh was a visiting professor at the Technion, Israel Institute of Technologyand a member at MSRI Berkeley. The hospitality of both institutions is gratefully acknowledged.2.
A Metrical approach
Our method is based on the following lemma in [13].
Lemma 2.1. [13, Lemma 1]
Let A and B be reals with B > A > . Let F N ( α ) be a nonnegativevalued function of N (an integer) and α (a real variable), and G N , V N functions of N such that:(i) G N → ∞ as N → ∞ ,(ii) V N = o ( G N ) as N → ∞ ,(iii) for all a , b with A ≤ a < b ≤ B we have lim sup N →∞ b Z a F N ( α ) G N dα ≥ b − a (iv) there is a positive constant K such that, for any measurable set C ⊆ [ A, B ] , Z C F N ( α ) dα ≤ KG N λ ( C ) + V N . Then, for almost all α ∈ [ A, B ] , we have lim sup N →∞ F N ( α ) G N ≥ . (5)In our application, F N ( α ) will be the number of solutions to (3) with p < N . Further, forgiven 0 < A < B we set G N = G N ( A, B ) = A B · min( c , ..., c d , d ) d − d +1 N − ( d +1)( γ d,k − ε ) (log N ) − We will prove
Theorem 2.2.
The following holds for every natural number N .(i) Let < A < B . Then for all a, b with A ≤ a < b ≤ B we have b Z a F N ( α ) dα ≥ ( b − a ) G N ( A, B )(1 + o (1)) (6) STEPHAN BAIER AND ANISH GHOSH if N ∈ S and N → ∞ .(ii) Let < A < B and ε > . Then there exists a constant K = K ( A, B, ε ) such that, for α ∈ [ A, B ] , we have F N ( α ) ≤ KG N ( A, B ) + J N ( α ) with B Z A | J N ( α ) | dα = o ( G N ( A, B )) as N → ∞ if N ∈ S and N → ∞ . Theorem 2.2(i) corresponds to Lemma 2, Theorem 2.2(ii) to Lemma 3 in [13]. Now it followsthat conditions (i) to (iv) in Lemma 2.1 are satisfied for V N = B Z A | J N ( α ) | dα. Now the claim in Theorem 1.1 follows from (5).3.
Proof of Theorem 2.2(i)
Reduction to a counting problem.
To prove Theorem 2.2(i), we broadly follow the ap-proach in section 3 of [13] . However, we use exponential sum estimates instead of zero densityestimates for the Riemann zeta function since they turn out to be more suitable for our purposes.This is the content of the next subsection.Throughout the sequel, we denote by P the set of primes. Let B p = [ r ∈ P q ,...,q d ∈ N (cid:20) rp , r + ˜ ηp (cid:19) ∩ (cid:20) c · q p , c · q + ˜ ηp (cid:19) ∩ · · · ∩ (cid:20) c d · q d p , c d · q d + ˜ ηp (cid:19) ∩ [ a, b ] , where ˜ η = p ε − γ d,k . Then b Z a F N ( α ) dα = X p ∈ P p ≤ N λ ( B p ) , (7)where λ is Lebesgue measure. Set µ := ( a + b ) / (2 a ) . (8)Our strategy is to split the interval [1 , N ] into subintervals [ P, P µ ] and sum up over the P ’s inthe end. Accordingly, we restrict p to the interval P ≤ p < P µ with P µ ≤ N . We then obtain alower bound for (7) by replacing ˜ η with η = ( µP ) ε − γ d,k . (9)Clearly, ˜ η ≥ η if P ≤ p < P µ .We note that if rp ≤ c i · q i p ≤ r + η/ p for i = 1 , ..., d, ESTRICTED SIMULTANEOUS DIOPHANTINE APPROXIMATION 5 then λ (cid:18)(cid:20) rp , r + ηp (cid:19) ∩ (cid:20) c · q p , c · q + ηp (cid:19) ∩ · · · ∩ (cid:20) c d · q d p , c d · q d + ηp (cid:19) (cid:19) ≥ ν, where ν := ηµP min (cid:18) , c , ..., c d (cid:19) = ( µP ) − − γ d,k + ε min (cid:18) , c , ..., c d (cid:19) . (10)Also, for all p ∈ [ P, P µ ), P aµ ≤ r ≤ bP = ⇒ a ≤ rp ≤ b, and r here runs over the primes in an interval of length b − a P . We thus have X P ≤ p<µP λ ( B p ) ≥ νN ( P ) , (11)where N ( P ) counts the number of solutions ( p, q , ..., q d , r ) ∈ P × Z d × P to q i ∈ [ c i r, c i r + δ ) for i = 1 , ..., d, P ≤ p < P µ, P aµ ≤ r ≤ bP, where δ := min { c , ..., c d } η { c , ..., c d } µP ) γ d,k − ε . (12)Note that in contrast to the problem considered by Harman and Jones, the conditions on p and q are here independent, which simplifies matters to some extent. By the prime number theorem,the number R ( P ) of prime solutions to P ≤ p < P µ satisfies R ( P ) ∼ ( µ − P (log 2 P ) − as P → ∞ . (13)It remains to count the number of solutions ( q , ..., q d , r ) ∈ N × P to q i ∈ [ c i r, c i r + δ ) for i = 1 , ..., d, P aµ ≤ r ≤ bP, which equals S ( P ) := X P aµ ≤ r ≤ bPr prime d Y i =1 ([ − c i r ] − [ − ( c i r + δ )]) . Let T ( P ) := X P aµ ≤ n ≤ bP d Y i =1 ([ − c i n ] − [ − ( c i n + δ )]) Λ( n ) . (14)We aim to show that T ( P ) = δ d ( b − aµ ) P (1 + o (1)) + O (cid:16) N − dγ d,k + ε/ (cid:17) if P µ ≤ N. (15)As usual, from (15), it follows that S ( P ) = δ d ( b − aµ ) P (log 2 P ) − (1 + o (1)) + O (cid:16) N − dγ d,k + ε/ (cid:17) if P µ ≤ N, which together with (13) gives N ( P ) = R ( P ) S ( P ) = δ d ( b − aµ )( µ − P (log 2 P ) − (1 + o (1)) + O (cid:16) P N − dγ d,k + ε/ (cid:17) if P µ ≤ N. STEPHAN BAIER AND ANISH GHOSH
Combing this with (8), (9), (10), (11) and (12), we obtain X P ≤ p<µP λ ( B p ) ≥ ( b − a ) a · min (2 , c , ..., c d ) d − d · ( µP ) − − ( d +1)( γ d,k − ε ) P (log 2 P ) − (1 + o (1))+ O (cid:16) P − γ d,k N − dγ d,k +3 ε/ (cid:17) if P µ ≤ N. (16)By splitting the interval [1 , N ) into intervals of the form [ P, µP ) and summing up, it nowfollows from (7) and (16) that b Z a F N ( α ) dα ≥ ( b − a ) a · min(2 , c , ..., c d ) d − d · ∞ X k =0 (cid:18) Nµ k (cid:19) − − ( d +1)( γ d,k − ε ) (cid:18) Nµ k +1 (cid:19) (log N ) − ! (1 + o (1))= ( b − a ) a · min(2 , c , ..., c d ) d − d · µ − · − µ − (1 − ( d +1)( γ d,k − ε )) · N − ( d +1)( γ d,k − ε ) (log N ) − (1 + o (1)) . Further, since µ >
1, we have1 − µ − (1 − ( d +1)( γ d,k − ε )) ≤ (1 − ( d + 1)( γ d,k − ε )) ( µ −
1) = (1 − ( d + 1)( γ d,k − ε )) · b − a a . Hence, we deduce that b Z a F N ( α ) dα ≥ ( b − a ) · a ( a + b ) · − ( d + 1)( γ d,k − ε ) · min ( c , ..., c d , d − d × N − ( d +1)( γ d,k − ε ) (log N ) − (1 + o (1)) ≥ ( b − a ) · A B · min ( c , ..., c d , d − d +1 · N − ( d +1)( γ d,k − ε ) (log N ) − (1 + o (1)) , (17)establishing the claim of Theorem 2.2(i). It remains to prove (15).3.2. Reduction to exponential sums.
For x ∈ R let ψ ( x ) := x − [ x ] − . Then we may write T ( P ) in the form T ( P ) = X P aµ ≤ n ≤ bP Λ( n ) d Y i =1 ( δ − ( ψ ( − c i n ) − ψ ( − ( c i n + δ ))))= X A⊆{ ,...,d } δ d −|A| T A ( P ) , (18) ESTRICTED SIMULTANEOUS DIOPHANTINE APPROXIMATION 7 where T A ( P ) := X P aµ ≤ n ≤ bP Λ( n ) Y i ∈A ( ψ ( − ( c i n + δ )) − ψ ( − c i n )) . By the prime number theorem, T ∅ ( P ) = δ d X P aµ ≤ n ≤ bP Λ( n ) ∼ δ d ( b − aµ ) P as P → ∞ . Hence, to establish (15), it suffices to prove that for any fixed ε > T A ( P ) = O (cid:16) N − dγ d,k + ε/ (cid:17) if P µ ≤ N (19)for all non-empty subsets A of { , ..., d } holds. We reduce the left-hand side to exponential sums,using the following Fourier analytic tool developed by Vaaler [20]. Lemma 3.1 (Vaaler) . For < | t | < let W ( t ) = πt (1 − | t | ) cot πt + | t | . Fix a positive integer J . For x ∈ R define ψ ∗ ( x ) := − X ≤| j |≤ J (2 πij ) − W (cid:18) jJ + 1 (cid:19) e ( jx ) and τ ( x ) := 12 J + 2 X | j |≤ J (cid:18) − | j | J + 1 (cid:19) e ( jx ) . Then τ ( x ) is non-negative, and we have | ψ ∗ ( x ) − ψ ( x ) | ≤ τ ( x ) for all real numbers x .Proof. This is Theorem A6 in [8] and has its origin in [20]. (cid:3)
We set τ ∗ ( x ) := ψ ( x ) − ψ ∗ ( x ) . Then T A ( P ) = X B⊆A X P aµ ≤ n ≤ bP Λ( n ) Y i ∈B ( ψ ∗ ( − ( c i n + δ )) − ψ ∗ ( − c i n )) ! × Y j ∈A\B ( τ ∗ ( − ( c j n + δ )) − τ ∗ ( − c j n )) = U A ( P ) + O ( V A ( P )) , (20)where U A ( P ) := X P aµ ≤ n ≤ bP Λ( n ) Y i ∈A ( ψ ∗ ( − ( c i n + δ )) − ψ ∗ ( − c i n )) ! and V A ( P ) := (log 2 P ) X i ∈A X P aµ ≤ n ≤ bP ( τ ( − ( c i n + δ )) + τ ( − c i n ))) , STEPHAN BAIER AND ANISH GHOSH where the O -term V A ( P ) arrives by using | ψ ∗ ( x ) | ≤ | τ ∗ ( x ) | ≤ τ ( x ) ≤
1, Λ( n ) ≤ log n and thetriangle inequality.The definition of the function τ ( x ) gives V A ( P ) = log 2 P J + 2 X i ∈A X | j |≤ J (cid:18) − | j | J + 1 (cid:19) (1 + e ( jδ )) X P aµ ≤ n ≤ bP e ( jc i n ) , and the definition of ψ ∗ ( x ) gives, after multiplying out and re-arranging summations, U A ( P ) = − X ≤| j |≤ J · · · X ≤| j h |≤ J h Y k =1 (cid:18) (2 πij k ) − W (cid:18) j k J + 1 (cid:19) (1 + e ( j k δ )) (cid:19)! × X P aµ ≤ n ≤ bP Λ( n ) e n X k ∈A j k c l k ! , where we suppose that J is a positive integer satisfying J ≤ P and h := |A| and A = { l , ..., l h } . We further estimate V A ( P ) by V A ( P ) ≪ P log 2 PJ + log 2 PJ · d X i =1 X ≤ j ≤ J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X P aµ ≤ n ≤ bP e ( jc i n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ P log 2 PJ + log 2 PJ · d X i =1 X ≤ j ≤ J min (cid:16) P, || jc i || − (cid:17) ≪ P log 2 PJ + (log 2 P ) d X i =1 X ≤ j ≤ J min (cid:18) Pj , || jc i || − (cid:19) =: P log 2 PJ + (log 2 P ) ˜ V d ( P ) , (21)where the term P (log 2 P ) /J bounds the contribution of j = 0, and we estimate U A ( P ) by U A ( P ) ≪ ˜ U A ( P ) := X ≤| j |≤ J · · · X ≤| j h |≤ J | j · · · j h | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X P aµ ≤ n ≤ bP Λ( n ) e n X k ∈A j k c l k !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (22)It remains to estimate ˜ U A ( P ) and ˜ V d ( P ). The estimation of ˜ V d ( P ) is clearly easier than that of˜ U A ( P ). We first deal with the term ˜ U A ( P ).3.3. Application of Vaughan’s identity.
We convert the inner sum involving the von Man-goldt function on the right-hand side of (22) into bilinear sums using Vaughan’s identity.
Lemma 3.2 (Vaughan) . Let u ≥ , v ≥ , uv ≤ x . Then we have for every arithmetic function f : N → C the estimate X u We use Lemma 3.2 with parameters u and v satisfying 1 ≤ u = v ≤ ( P aµ ) / , to be fixed later, x := bP and f ( n ) := ( e ( nc ) if P aµ ≤ n ≤ bP, n < P aµ with c := X k ∈A j k c l k to deduce that ˜ U A ( P ) ≪ (log 2 P ) Z + Z , (23)where Z := X ≤| j |≤ J · · · X ≤| j h |≤ J | j · · · j h | X l ≤ u max P aµ/l ≤ w ≤ bP/l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X w ≤ m ≤ bP/l e ml X k ∈A j k c l k !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and Z := X ≤| j |≤ J · · · X ≤| j h |≤ J | j · · · j h | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X u Exchanging summation and estimating geometric sums, we deduce that Z ( L ) ≪ LP (log 2 P ) h +5 + L (log 2 P ) h +3 X ≤| j |≤ J · · · X ≤| j h |≤ J | j · · · j h | × X u ≤ m The treatments of Z and Z lead to sumsof the form R A ( M, x ) := X ≤| j |≤ J · · · X ≤| j h |≤ J | j · · · j h | X ≤ m ≤ M min xm , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X k ∈A j k c l k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (27)which we estimate in the following. Here the Diophantine properties of the vector ( c , ..., c d ) comeinto play.Splitting each of the j k -intervals into O (log P ) dyadic intervals and summing up their contri-butions, we obtain R A ( M, x ) ≪ (log 2 P ) d sup ≤ H ,...,H d ≤ J R d ( H , ..., H d , M, x ) H · · · H d , (28)where R d ( H , ..., H d , M, x ) := X | j |≤ H ,..., | j d |≤ H d j = X ≤ m ≤ M min (cid:16) xm , || m j · c || − (cid:17) (29)with j := ( j , ..., j d ) and c := ( c , ..., c d ) . The key point is now to approximate j · c by rational numbers, using Dirichlet’s approximationtheorem, where the denominators are uniformly bounded from below. Let j ∈ Z d \ and X be apositive integer, to be fixed later. By Dirichlet’s approximation theorem, there exist integers a, q such that gcd( a, q ) = 1, 1 ≤ q ≤ X and (cid:12)(cid:12)(cid:12)(cid:12) j · c − aq (cid:12)(cid:12)(cid:12)(cid:12) ≤ qX . Hence, || q j · c || ≤ X . On the other hand, by condition (1) in Theorem 1.1, we have Cq k || j || k ∞ < || q j · c || . It follows that q > ( CX ) /k || j || ∞ . Now we apply the following well-known lemma. Lemma 3.3. Let L ≥ and x > . Suppose that | c − a/q | ≤ q − with a ∈ Z , q ∈ N and ( a, q ) = 1 .Then X ≤ l ≤ L min (cid:16) xl , || lc || − (cid:17) ≪ (cid:18) xq + L + q (cid:19) (log 2 Lqx ) . Proof. This is Lemma 6.4.4. in [3]. (cid:3) It follows that X ≤ m ≤ M min (cid:16) xm , || m j · c || − (cid:17) ≪ (cid:18) x || j || ∞ X /k + M + X (cid:19) (log 2 M Xx )and hence R d ( H , ..., H d , M, x ) ≪ H · · · H d (cid:18) x max( H , ..., H d ) X /k + M + X (cid:19) (log 2 M Xx ) . (30)Combining (28) and (30), we obtain R A ( M, x ) ≪ (log 2 P ) d +1 (cid:18) xJX /k + M + X (cid:19) . Now choosing X := h ( xJ ) − / ( k +1) i , we deduce that R A ( M, x ) ≪ (log 2 P ) d +1 (cid:16) M + ( xJ ) − / ( k +1) (cid:17) . (31)3.5. Completion of the proof. Using (24), (26), (28) and (31), we get Z ≪ (log 2 P ) d +1 (cid:16) u + ( P J ) − / ( k +1) (cid:17) (32)and Z ( L ) ≪ (log 2 P ) d/ / (cid:16) P / L / + P L − / + P − / (2( k +1)) J / − / (2( k +1)) (cid:17) and hence by (25), Z ≪ (log 2 P ) d/ / (cid:16) P u − / + P − / (2( k +1) J / − / (2( k +1)) (cid:17) . (33)Combining (22), (23), (32) and (33), we get˜ U A ( P ) ≪ P ε (cid:16) u + P u − / + ( P J ) − / ( k +1) + P − / (2( k +1)) J / − / (2( k +1)) (cid:17) . ESTRICTED SIMULTANEOUS DIOPHANTINE APPROXIMATION 13 Now choosing u := P / , it follows that˜ U A ( P ) ≪ P ε (cid:16) P / + ( P J ) − / ( k +1) + P − / (2( k +1)) J / − / (2( k +1)) (cid:17) . (34)We now turn to the term ˜ V d ( P ), defined in (21). By (29) and (30), we have˜ V d ( P ) ≪ R d (1 , ..., , J, P ) ≪ (cid:18) PX /k + J + X (cid:19) (log 2 J XP )for any positive integer X . Choosing X := h P − / ( k +1) i gives ˜ V d ( P ) ≪ log(2 P ) (cid:16) J + P − / ( k +1) (cid:17) . Combining this with (21), we find V A ( P ) ≪ (log 2 P ) (cid:16) P J − + J + P − / ( k +1) (cid:17) . (35)Now putting (20), (34) and (35) together, we arrive at T A ( P ) ≪ P ε (cid:16) P J − + J + P / + ( P J ) − / ( k +1) + P − / (2( k +1)) J / − / (2( k +1)) (cid:17) . Now choosing J := h P / (3 k +2) i , (36)we get T A ( P ) ≪ P − / (3 k +2)+ ε , which proves (19) upon replacing ε by ε/ 2. This completes the proof of Theorem 2.2(i).4. Proof of Theorem 2.2(ii) Sieve theoretical approach. We are broadly following the treatment in [13] with appro-priate modifications because the linear case, considered here, requires a different treatment. Inparticular, as in the previous section, the Diophantine properties of the vector ( c , ..., c d ) willcome into play. Let {·} represent the fractional part, and put µ := N ε − γ d,k . Write A = A ( α ) = { n [ nα ] : 1 ≤ n ≤ N, { nα } < µ, { nc i α } < µ for i = 1 , ..., d } . We desire to show that A does not contain too many products of two primes. To this end, weapply a two-dimensional upper bound sieve (see [9], Theorem 5.2). We therefore need to obtainan asymptotic formula for the number of solutions to n [ nα ] ≡ q, ≤ n ≤ N, with { nα } < µ and { nc i α } < µ for i = 1 , ..., d, (37)where q ≤ Q := N ε . For this it suffices to establish a formula for the number of solutions to n ≡ t , [ nα ] ≡ t subject to (37). We can combine (37) with the congruence conditions to require1 ≤ n ≤ Nt , (cid:26) nt αt (cid:27) < µt , { nt c i α } < µ for i = 1 , ..., d, (38)and count the number S ( α ; t , t ) of solutions to (38) using Fourier analysis. To this end, wewrite S ( α, t , t ) = X ≤ n ≤ N/t (cid:18)(cid:20) nt αt (cid:21) − (cid:20) nt αt − µt (cid:21)(cid:19) · d Y i =1 ([ nt c i α ] − [ nt c i α − µ ])and evaluate this term in a similar way as the term T ( P ) defined in (14) using Vaaler’s Lemma3.1. By a chain of similar calculations, we arrive at the asymptotic estimate S ( α ; t , t ) = N µ d +1 t t + O (cid:18) N µ d L + E ( α ; t , t ) (cid:19) , (39)where we set L := Q µ − and E ( α ; t , t ) := µ d +1 t X | m |≤ L,..., | m d |≤ L ( m ,...,m d ) =(0 ,..., (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ≤ n ≤ N/t e nαt m t + d X i =1 c i m i !!(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The above estimate (39) is analog to the equation after (26) in [13].Now, applying the upper bound sieve gives F N ( α ) ≤ C ( ε ) N µ d log N + O ( J N ( α )) , (40)where J N ( α ) := X t t ≤ Q ( t t ) ε (cid:18) N µ d L + E ( α ; t , t ) (cid:19) = X t t ≤ Q ( t t ) ε E ( α ; t , t ) + o (cid:18) N µ d +1 log N (cid:19) as N → ∞ . Hence, to establish the claim in Theorem 2.2(i), it suffices to show that X t t ≤ Q ( t t ) ε B Z A E ( α ; t , t ) dα = o (cid:18) N µ d +1 log N (cid:19) as N → ∞ , N ∈ S . (41) ESTRICTED SIMULTANEOUS DIOPHANTINE APPROXIMATION 15 Average estimation for E ( α ; t , t ) . To estimate the expression on the right-hand side of(41), we first observe that E ( α ; t , t ) ≪ µ d +1 t X | m |≤ L,..., | m d |≤ L ( m ,...,m d ) =(0 ,..., min Nt , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) αt (cid:18) m t + m c + ... + m d c d (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ! (42)and note that if ( m , ..., m d ) = (0 , ..., t (cid:18) m t + m c + ... + m d c d (cid:19) on the right-hand side of (42) is non-zero because ( c , ..., c d ) is k -Diophantine. For an estimationof the integral in (41), we now use the following lemma. Lemma 4.1. Suppose that < A < B , K ≥ and x = 0 . Then B Z A min (cid:16) K, || αx || − (cid:17) dα = O A,B (cid:0) min (cid:8) K, max (cid:8) , | x | − (cid:9)(cid:9) log K (cid:1) . (43) Proof. We confine ourselves to the case when x > x < β = αx , we get B Z A min (cid:16) K, || αx || − (cid:17) dα = 1 x xB Z xA min (cid:16) K, || β || − (cid:17) dβ. (44)By periodicity of the integrand, if x ( B − A ) ≥ 1, we have1 x xB Z xA min (cid:16) K, || β || − (cid:17) dβ ≤ [ x ( B − A ) + 1] x Z min (cid:16) K, || β || − (cid:17) dβ = O A,B (log K ) . (45)If 1 /K ≤ x ( B − A ) < 1, then1 x xB Z xA min (cid:16) K, || β || − (cid:17) dβ ≤ x x ( B − A ) / Z − x ( B − A ) / min (cid:16) K, || β || − (cid:17) dβ = O A,B (cid:18) log Kx (cid:19) . (46)If 0 < x ( B − A ) < /K , then trivially1 x xB Z xA min (cid:16) K, || β || − (cid:17) dβ = O A,B ( K ) . (47)Combining (44), (45), (46) and (47), we deduce the claim when x > 0, which completes theproof. (cid:3) Now, employing (42) and Lemma 4.1, we get B Z A E ( α ; t , t ) dα ≪ µ d +1 t X | m |≤ L,..., | m d |≤ L ( m ,...,m d ) =(0 ,..., min ( Nt , max ( , (cid:12)(cid:12)(cid:12)(cid:12) t (cid:18) m t + m c + ... + m d c d (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) − ) log N ) ≪ µ d +1 (log N ) X | m |≤ L,..., | m d |≤ L ( m ,...,m d ) =(0 ,..., min (cid:26) Nt t , max (cid:26) t , | t m + t t ( m c + ... + m d c d ) | − (cid:27)(cid:27) . The contribution of ( m , ...m d ) = (0 , ..., 0) to the last line is bounded by O (cid:0) µ d +1 L log N (cid:1) , andthe contribution of ( m , ..., m d ) = 0 is bounded by ≪ µ d +1 (log N ) X | m |≤ L,..., | m d |≤ L ( m ,...,m d ) =(0 ,..., X | m |≤ L min (cid:26) Nt t , max (cid:26) t , | t m + t t ( m c + ... + m d c d ) | − (cid:27)(cid:27) ≪ µ d +1 (log N ) X | m |≤ L,..., | m d |≤ L ( m ,...,m d ) =(0 ,..., min (cid:26) Nt t , || t t ( m c + ... + m d c d ) || − (cid:27) + µ d +1 L d +1 log N. So altogether, we have B Z A E ( α ; t , t ) dα ≪ µ d +1 L d +1 log N + µ d +1 (log N ) X | m |≤ L,..., | m d |≤ L ( m ,...,m d ) =(0 ,..., min (cid:26) Nt t , || t t ( m c + ... + m d c d ) || − (cid:27) . Summing up over t and t , writing q = t t and using d ( q ) = O ( q ε ), d ( q ) being the divisorfunction, we obtain X t t ≤ Q ( t t ) ε B Z A E ( α ; t , t ) dα ≪ N ε QL d +1 µ d +1 + N ε µ d +1 R d ( L, ..., L, Q, N ) , where R d ( L, ..., L, Q, N ) is defined as in (29). Combining this with (30), we get X t t ≤ Q ( t t ) ε B Z A E ( α ; t , t ) dα ≪ N ε L d µ d +1 (cid:18) L + LNX /k + Q + X (cid:19) log(2 QN X )for any positive integer X . Choosing X := (cid:2) ( LN ) − / ( k +1) (cid:3) , we deduce that X t t ≤ Q ( t t ) ε B Z A E ( α ; t , t ) dα ≪ N ε L d µ d +1 (cid:16) L + Q + ( LN ) − / ( k +1) (cid:17) ESTRICTED SIMULTANEOUS DIOPHANTINE APPROXIMATION 17 which implies (41) upon recalling µ := N ε − γ d,k , Q := N ε , L := Q µ − and (2). References [1] V. Beresnevich, V. 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