Badly approximable numbers and Littlewood-type problems

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1 Math. Proc. Camb. Phil. Soc.: page of 2 c Cambridge Philosophical Society 20 doi:0.07/s Badly approximable numbers and Littlewood-type problems BY YANN BUGEAUD Université de Strasbourg, Mathématiues, 7, rue René Descartes, Strasbourg, Cedex, France. bugeaud@math.unistra.fr AND NIKOLAY MOSHCHEVITIN Moscow State University, Number Theory, Leninskie Gory, Moscow, Russian Federation. moshchevitin@rambler.ru (Received 7 May 2009; revised 2 September 200) Abstract We establish that the set of pairs (α, β) of real numbers such that lim inf + (log )2 α β > 0, where denotes the distance to the nearest integer, has full Hausdorff dimension in R 2. Our proof rests on a method introduced by Peres and Schlag, that we further apply to various Littlewood-type problems.. Introduction A famous open problem in simultaneous Diophantine approximation, called the Littlewood conjecture [5], claims that, for any given pair (α, β) of real numbers, we have inf α β =0, ( ) where denotes the distance to the nearest integer. Throughout the present paper, we denote by Bad the set of badly approximable numbers, that is, Bad ={α R : inf α > 0}, and we recall that Bad has Lebesgue measure zero and full Hausdorff dimension []. Conseuently, ( ) holds for almost every pair (α, β) of real numbers. Recently, this result was considerably improved by Einsiedler, Katok and Lindenstrauss [9], who established that the set of pairs (α, β) for which ( ) do not hold has Hausdorff dimension zero; see also [25] for a weaker statement, and [5, section 0 ] for a survey of related results. Another metrical statement connected to the Littlewood conjecture was established by Gallagher [2] in 962 and can be formulated as follows (see e.g. []). THEOREM G. Let n be a positive integer. Let : R >0 R >0 be a non-increasing function. The set of points (x,...,x n ) in R n such that there are infinitely many positive integers

2 2 YANN BUGEAUD AND NIKOLAY MOSHCHEVITIN satisfying n x i < () has full Lebesgue measure if the sum (h) n (log h) n i= h diverges, and has zero Lebesgue measure otherwise. In particular, it follows from Gallagher s theorem that lim inf (log + )2 α β =0 ( 2) for almost every pair (α, β) of real numbers. The main purposes of the present note are to establish the existence of exceptional pairs (α, β) which do not satisfy ( 2) a result first provedin[22], and to prove that the set of these pairs has full Hausdorff dimension in R 2. We further consider various uestions closely related to the Littlewood conjecture. Our main results are stated in Section 2 and proved in Sections 4 and 5, with the help of auxiliary lemmas gathered in Section. Several additional results are given without proofs in Section 6. Throughout this paper, x and x denote the greatest integer less than or eual to x and the smallest integer greater than or eual to x, respectively. 2. Main results Our first result shows that there are many pairs (α, β) of real numbers that are not well multiplicatively approximable. THEOREM. For every real number α in Bad, the set of real numbers β such that lim inf (log + )2 α β > 0 (2 ) has full Hausdorff dimension. The proof of Theorem uses a method introduced by Peres and Schlag [24], which was subseuently applied in [8 22]. Since the set Bad has full Hausdorff dimension, the next result follows from Theorem by an immediate application of [, corollary 7 2]. THEOREM 2. The set of pairs (α, β) of real numbers satisfying lim inf (log + )2 α β > 0 has full Hausdorff dimension in R 2. Theorem can be viewed as a complement to the following result of Pollington and Velani [25]. THEOREM PV. For every real number α in Bad, there exists a subset G(α) of Bad with full Hausdorff dimension such that, for any β in G(α), there exist arbitrarily large integers satisfying (log ) α β.

3 Badly approximable numbers and Littlewood-type problems In [], the authors constructed explicitly for every α in Bad uncountably many β in Bad such that the pair (α, β) satisfies ( ), and even a strong form of this ineuality. It would be very interesting to construct explicit examples of pairs of real numbers that satisfy (2 ). A modification of an auxiliary lemma yields a slight improvement on Theorem. THEOREM. Let a be a real number with 0 < a <. For every real number α in Bad, the set of real numbers β such that lim inf (log + )2 a (log / α ) a α β > 0 has full Hausdorff dimension. Theorem is stronger than Theorem since, for every α in Bad, there exists a positive real number δ such that log(/ α ) δ log holds for every integer 2. Cassels and Swinnerton Dyer [8] proved that ( ) is euivalent to the euality inf max{ x, } max{ y, } xα + yβ =0, (x,y) Z Z\{(0,0)} and used it to show that ( ) holds if α and β belong to the same cubic number field (see also [2]). In this context, we have the following metrical result, extracted from [4, page 455]. For integers,..., n, set n (,..., n ) = max{, i }. THEOREM BKM. Let n be a positive integer. Let : R >0 R >0 be a non-increasing function. The set of points (x,...,x n ) in R n such that there are infinitely many integers,..., n satisfying has full Lebesgue measure if the sum i= x + + n x n < ( (,..., n ) ) (2 2) h diverges, and has zero Lebesgue measure otherwise. (h)(log h) n (2 ) For n 2, there is no known example of points (x,...,x n ) in R n and of a function as in Theorem BKM such that the sum (2 ) diverges and (2 2) has only finitely many solutions. The Peres Schlag method allows us to show that such examples do exist. THEOREM 4. The set of pairs (α, β) of real numbers satisfying lim inf x,y 0 max{2, xy } xα + yβ (log max{2, xy }) 2 > 0 has full Hausdorff dimension in R 2. The proof of Theorem 4 is briefly outlined in Section 5. Note that Theorem 4 (resp. Theorem ) does not follow from Theorem (resp. Theorem 4) by some transference principle. In analogy with the Littlewood conjecture, de Mathan and Teulié [7] proposed recently a mixed Littlewood conjecture. For any prime number p, the usual p-adic absolute value p is normalized in such a way that p p = p.

4 4 YANN BUGEAUD AND NIKOLAY MOSHCHEVITIN De Mathan Teulié conjecture. For every real number α and every prime number p, we have inf α p = 0. Despite several recent results [6, 0], this conjecture is still unsolved. The following metrical statement, established in [7], should be compared with Theorem G. THEOREM BHV. Let k be a positive integer. Let p,...,p k be distinct prime numbers. Let : R >0 R >0 be a non-increasing function. The set of real numbers α such that there are infinitely many positive integers satisfying has full Lebesgue measure if the sum α p pk < () h (h)(log h) k diverges, and has zero Lebesgue measure otherwise. As an immediate conseuence of Theorem BHV, we get that, for every prime number p, almost every real number α satisfies inf (log 2 )2 (log log ) α p = 0. (2 4) The method of proof of Theorem allows us to confirm the existence of real numbers for which (2 4) does not hold. THEOREM 5. Let a be a real number with 0 a <. For every prime number p, the set of real numbers α such that lim inf (log + )2 a α p (log 2/ p ) a > 0 has full Hausdorff dimension. We display an immediate conseuence of Theorem 5. COROLLARY. For every prime number p, the set of real numbers α such that lim inf (log + )2 α p > 0 has full Hausdorff dimension. In the present note, we have restricted our attention to 2-dimensional uestions. However, our method can be successfully applied to prove that, given an integer n 2, there are real numbers α,...,α n such that lim inf (log + )n α α n > 0, as well as real numbers β,...,β n such that lim inf max{2, x...x n } x β x n β n (log max{2, x...x n }) n > 0 x,...,x n 0 This will be the subject of subseuent work by E. Ivanova.

5 Badly approximable numbers and Littlewood-type problems 5. Auxiliary results The original method of Peres and Schlag is a construction of nested intervals. A useful tool for estimating from below the Hausdorff measure of a Cantor set is the mass distribution principle, which we recall now. We consider a set K included in a bounded interval E, and defined as follows. Set E 0 = E and assume that, for any positive integer k, there exists a finite family E k of disjoint compact intervals in E such that any interval U belonging to E k is contained in exactly one of the intervals of E k and contains at least two intervals belonging to E k+. Suppose also that the maximum of the lengths of the intervals in E k tends to 0 when k tends to infinity. For k 0, denote by E k the union of the intervals belonging to the family E k, and set K := + k= LEMMA. Keep the same notation as above. Assume further that there exists a positive integer k 0 such that, for any k k 0, each interval of E k contains at least m k 2 intervals of E k, these being separated by at least k, where 0 < k+ < k. We then have dim K lim inf k + E k. log(m...m k ). log(m k k ) Proof. This is [, example 4 6], see also [5, proposition 5 2]. LEMMA 2. Let α be in Bad. There exists a positive constant C(α) such that, for every integer 2, we have x= C(α). αx x log 2 2 x Proof. This is a straightforward conseuence of [4, example 2, page 24], where it is established that there exists a positive constant C (α) such that m αx x C (α)(log m) 2, for all positive integers m. x= Theorem depends on the following refinement of Lemma 2. LEMMA. Let α be in Bad. Let a be a real number with 0 < a <. There exists a positive constant C(α) such that, for every integer 2, we have x= C(α). αx x (log / xα ) a (log x) 2 a Proof. Let (p j / j ) j 0 denote the seuence of convergents to α. Let m (resp. n) be the largest (resp. the smallest) integer j such that j (resp. j ). As the seuence ( j ) j 0 grows exponentially fast, we have log n m log, where, as throughout this proof, the numerical constants implied by depend only on α.

6 6 YANN BUGEAUD AND NIKOLAY MOSHCHEVITIN Let j be an integer satisfying m j < n and consider S j := j+ αx x (log / xα ). a x= j Denote by y < y 2 < < y j+ the increasing seuence of the j+ points {αx}, x =,..., j+. It follows from the Three Distance Theorem (see, e.g. [2, section ]) and the fact that α is badly approximable that x y x x, for x =,..., j+. j+ j+ Conseuently, S j j j+ x= αx (log / xα ) a j j+ j j+ /2 x= j+ 2 ( j+ /x) (log( j+ /x)) a du u(log u) a (log j) a, since j+ / j is bounded from above by an absolute constant depending only on α. Now, which proves the lemma. n S j (log ) 2 a, j=m The key tool for the proof of Theorem 5 is Lemma 4 below. LEMMA 4. Let p be a prime number. Let a be a real number with 0 a <. There exists a positive constant C(a, p) such that, for every integer 2, we have x= C(a, p). x x p (log(2/ x p )) a (log x) 2 a Proof. Observe that Conseuently, x= log x x p (log(2/ x p )) a j=0 /p j x= /p j x( j + ) a. x= and the lemma is proved. log x x p (log(2/ x p )) a j=0 log ( j + ) a (log )2 a, 4. Proof of Theorem Let α be in Bad and δ be a positive real number satisfying α δ, for every. (4 )

7 Let be such that Badly approximable numbers and Littlewood-type problems 7 0 << ( 2 0 C(α) ), (4 2) where C(α) is given by Lemma 2. We follow a method introduced by Peres and Schlag [24]. First, we construct dangerous sets of real numbers. These sets depend on α, but, to simplify the notation, we choose not to indicate this dependence. For integers x and y with x 2 and 0 y x, define [ y E(x, y) = x αx x 2 log 2 2 x, y ] x + αx x 2 log 2 2 x (4 ) and x ( ) E(x) = E(x, y) [0, ]. (4 4) y=0 Set also l 0 = 0, l x = log 2 ( αx x 2 log 2 2 x/(2)), for x Z. (4 5) Each interval from the union E(x) defined in (4 4) can be covered by an open dyadic interval of the form ( b, b + 2 ), b Z 2 l x 2 l 0. x Let A(x) be the smallest union of all such dyadic intervals which covers the whole set E(x) and put A c (x) =[0, ]\A(x). Observe that A c (x) is a union of closed intervals of the form [ a, a + ], a Z 2 l x 2 l 0. x Let 0 be an integer such that 0 (00) and 0 α /4. (4 6) For 0, define B = A c (x). x= 0 The sets B, 0, are closed and nested. Our aim is to show inductively that they are non-empty. Set L 0 = l 0 and k := k 0, L k = log 2 ( 2 k log 2 2 k/(4)), k. (4 7) Observe that l x L k when x k. For every integer k 0 we construct inductively subsets C k and D k of B k with the following property (P k ): The set C k is the union of 2 5k +L k intervals of length 2 L k, separated by at least 2 L k, and such that at least 2 5k 5+L k among them include at least 2L k+ L k intervals composing B k+, which are also separated by at least 2 L k+. Let denote by C k+ (resp. by D k ) the

8 8 YANN BUGEAUD AND NIKOLAY MOSHCHEVITIN union of 2 5(k+) +L k+ of these intervals (resp. of the corresponding 2 5k 5+L k intervals from C k ). In particular, we have mes(c k ) = 4mes(D k ) = 2 5 mes(c k+ ). We deduce from (4 2), (4 ) and Lemma 2 that mes(b ) mes ( A(x) ) /2. x= 0 Conseuently, B is the union of at least 2 L intervals of length 2 L.By(4 6), the set B 0 is the union of at least 2 L0 intervals of length 2 L 0. This allows us to define the sets C 0, D 0 and C. This proves (P 0 ). Let k be a non-negative integer such that (P k ) holds, and consider the set B k+2 := C k+ B k+2. Observe that ( k+2 ) B k+2 = C k+ \ A(x), hence mes(b k+2 ) mes(c k+ ) x= k+ + k+2 x= k+ + By construction, the set C k can be written as a union, say of T k dyadic intervals J ν of the form [ a C k =, a + 2 L k 2 L k T k ν= J ν, ], a Z 0, where L k is given by (4 7). Let x k be an integer. Since, by (4 6), 2 L k 2 k log 2 2 k 4 mes ( C k+ A(x) ). (4 8) k 2 x 2, each interval J ν contains at least the rationals y/x,(y + )/x for some integer y, and we infer from (4 ) that 2 4 mes(j ν A(x)) αx x log 2 2 x mes(j ν). (4 9) Summing (4 9) from ν = toν = T k, we get It then follows from (4 0) that mes ( C k A(x) ) mes(c k+ A(x)) mes(c k A(x)) 2 4 αx x log 2 2 x mes(c k ) 2 4 αx x log 2 2 x mes(c k ). (4 0) 2 9 αx x log 2 2 x mes(c k+ ).

9 Badly approximable numbers and Littlewood-type problems 9 Combined with (4 8) and Lemma 2, this gives mes(b k+2 ) ( mes(c k+ ) ) ( k+2 x= k αx x log 2 2 x ) mes(c k+ ). 2 Thus, at least one uarter of the intervals composing C k+ contains at least 2 L k+2 L k+ 2 intervals composing B k+2, thus at least 2 L k+2 L k+ intervals composing B k+2, if we impose that these intervals are mutually distant by at least 2 L k+2. This allows us to define the sets C k+2 and D k+ with the reuired properties. This proves (P k+ ). It then follows that the set K := k 0 D k is non-empty. By construction, every point β in this set avoids all the intervals E(x, y) with x 0, thus, the pair (α, β) satisfies (2 ). To establish that the set K has full Hausdorff dimension, we apply Lemma with m k = 2 L k+ L k 5 Note that log(m...m k ) log(m k k ) We infer from (4 ), (4 5) and (4 7) that and k := 2 L k+. log(2 k 2 L k ) log(2 L k+5 ) Conseuently, lim k + 2 L k δ k 0. log(m...m k ) log(m k k ) =, and it follows from Lemma that the set K has full Hausdorff dimension. This completes the proof of our theorem. 5. Proofs of Theorems, 4 and 5 The proofs of Theorems and 5 follow exactly the same steps as that of Theorem. Instead of the intervals [ y E(x, y) = x αx x 2 log 2 2 x, y ] x + αx x 2 log 2 2 x, we use respectively the intervals [ y x αx x 2 (log 2 x) 2 a (log / αx ), y ] a x + αx x 2 (log 2 x) 2 a (log / αx ) a and [ y x x p x 2 (log 2 x) 2 a (log 2/ x p ), y a x + Furthermore, we apply Lemmas and 4 in place of Lemma 2. x p x 2 (log 2 x) 2 a (log 2/ x p ) a ].

10 0 YANN BUGEAUD AND NIKOLAY MOSHCHEVITIN For the proof of Theorem 4, we work directly in the plane. The idea is the following. For a triple (x, y, z) of integers and a positive, the ineuality xx+ yy + z defines a strip composed of points (X, Y ) close to the line xx + yy + z = 0. Since we are working in the unit suare, to a given pair (x, y) of integers corresponds a uniue z, and the length of the intersection of the line with the unit suare is at most eual to 2. Setting x,y = xy log 2 xy, for a given (very small) positive, the strips xx+ yy + z x,y play the same role as the intervals (4 ) in the proof of Theorem. Since, for every large integer, wehave xy x,y x= x= /x y= /x xy log 2, x log the Peres Schlag method can be applied as in the proof of Theorem. We omit the details. 6. Further results We gather in the present section several results that can be obtained with the same method as in the proof of Theorem, that is, by combining the Peres Schlag method with the mass distribution principle. A result on lacunary seuences. THEOREM 6. Let M be a positive real number and (t j ) j be a seuence such that t j+ /t j > + /M for j. Let c be a real number with 0 < c < /0. Let be a positive real number. Then, the Hausdorff dimension of the set is at least if M is sufficiently large. {ξ [0, ] : n, ξt n c/(m log M)} Theorem 6 complements the results from [8, 24]. A result on seuences with polynomial growth. THEOREM 7. Let C, C 2 and γ be positive real numbers. Let (t n ) n be a seuence of real numbers such that C n γ t n C 2 n γ, for n. Then, there exist a positive C and an integer n 0 such that the set { ξ R : ξt n > C } n log n has full Hausdorff dimension. n n 0 (6 )

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