METRICAL RESULTS ON THE DISTRIBUTION OF FRACTIONAL PARTS OF POWERS OF REAL NUMBERS

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1 METRICAL RESULTS ON THE DISTRIBUTION OF FRACTIONAL PARTS OF POWERS OF REAL NUMBERS YANN BUGEAUD, LINGMIN LIAO, AND MICHA L RAMS Abstract Denote by { } the fractional part We establish several new metrical results on the distribution properties of the sequence ({x n }) n Many of them are presented in a more general framework, in which the sequence of functions (x x n ) n is replaced by a sequence (f n) n, under some growth and regularity conditions on the functions f n Introduction Let { } denote the fractional part and the distance to the nearest integer For a given real number x >, only few results are known on the distribution of the sequence ({x n }) n For example, we still do not know whether 0 is a limit point of ({e n }) n, nor of ({( 3 2 )n }) n ; see [5] for a survey of related results However, several metric statements have been established The first one was obtained in 935 by Koksma [3], who proved that for almost every x > the sequence ({x n }) n is uniformly distributed on the unit interval [0, ] Here and below, almost every always refers to the Lebesgue measure In 967, Mahler and Szekeres [5] studied the quantity P (x) := lim inf x n /n (x > ) They proved that if P (x) = 0 then x is transcendental, and P (x) = for almost all x > The function x P (x) was subsequently studied in 2008 by Bugeaud and Dubickas [6] Among other results, it was shown in [6] that, for all v > u > and b >, we have dim H {x (u, v) : P (x) /b} = log v log(bv), where dim H denotes the Hausdorff dimension In a different direction, Pollington [6] showed in 980 that there are many real numbers x > such that ({x n }) n is very far from being well distributed, namely he established that, for any ε > 0, we have dim H { x > : {x n } < ε for all n } = This result has been subsequently extended by Bugeaud and Moshchevitin [8] and, independently, by Kahane [], who proved that for any ε > 0, for any sequence of real numbers (y n ) n, we have dim H { x > : x n y n < ε for all n } = 200 Mathematics Subject Classification: Primary K36 Secondary J7, 28A80 Key words and phrases: fractional parts of powers, Diophantine approximation, Hausdorff dimension M R was supported by National Science Centre grant 204/3/B/ST/0033 (Poland)

2 2 YANN BUGEAUD, LINGMIN LIAO, AND MICHA L RAMS In the present paper, we further investigate, from a metric point of view, the Diophantine approximation properties of the sequence ({x n }) n, where x >, and extend several known results to more general families of sequences ({f n (x)}) n, under some conditions on the sequence of functions (f n ) n As a consequence of our main theorem, we obtain an inhomogeneous version of the result of Bugeaud and Dubickas [6] mentioned above Theorem Let b > be a real number and y = (y n ) n an arbitrary sequence of real numbers in [0, ] Set E(b, y) := {x > : x n y n < b n for infinitely many n} For every v >, we have lim dim H([v ε, v + ε] E(b, y)) = ε 0 log v log(bv) In the homogeneous case (that is, the case where y n = 0 for n ), Theorem was proved in [6] by using a classical result of Koksma [4] and the mass transference principle developed by Beresnevich and Velani [3] The method of [6] still works when y is a constant sequence, but one then needs to apply the inhomogeneous version of Koksma s theorem in [4] Here, for an arbitrary sequence (y n ) n, we use a direct construction Letting v tend to infinity in Theorem, we obtain the following immediate corollary Corollary 2 For an arbitrary sequence y of real numbers in [0, ] and any real number b >, the set E(b, y) has full Hausdorff dimension Theorem gives, for every v >, the value of the localized Hausdorff dimension of E(b, y) at the point v We stress that, in the present context, the localized Hausdorff dimension varies with v, while this is not at all the case for many classical results, including the Jarník Besicovitch Theorem and its extensions Taking this point of view allows us also to place Theorem in a more general context, where the family of functions x x n is replaced by an arbitrary family of functions f n satisfying some regularity and growth conditions We consider a family of strictly positive increasing C functions f = (f n ) n defined on an open interval I R and such that f n (x), f n(x) > for all x I For τ >, define E(f, y, τ) := {x I : f n (x) y n < f n (x) τ for infinitely many n} For v I, put u(v) := lim sup log f n (v) log f n(v), We will assume the regularity condition () lim lim sup r 0 sup x y <r log f n (v) l(v) := lim inf log f n(v) log f n(x) log f n(y) =, which guarantees the continuity of the functions u and l

3 DISTRIBUTION OF FRACTIONAL PARTS 3 For non-linear functions f n, ie, when f n is not of the form f n (x) = a n x + b n, we also need the following condition: log f n+ (2) M := sup (v) n log f n(v) < for all v I Theorem is a particular case of the following general statement Theorem 3 Consider a family of strictly positive increasing C functions f = (f n ) n defined on an open interval I R and such that f n (x), f n(x) > for all x I Assume () and (2) If for all x I, (3) ε > 0, f n(x) ε <, n= then, for any v I and any τ >, we have + τu(v) lim ε 0 dim H([v ε, v + ε] E(f, y, τ)) + τl(v) If the functions f n are linear then we do not need to assume (2), and the assertion gets strengthened to (4) lim dim H([v ε, v + ε] E(f, y, τ)) = ε 0 We remark that the condition (3) is satisfied if x I, log f lim n(x) = log n + τl(v) We also observe that the condition () implies that l(v) for v in I In many cases (in particular, for f n (x) = x n ), we have u(v) = l(v) = for v in I It follows from the formulation of Theorem 3 that the real number τ can be replaced by a continuous function τ : I (0, ), in which case the set E(f, y, τ) is defined by E(f, y, τ) := {x I : f n (x) y n < f n (x) τ(x) for infinitely many n} We get at once the following localized version of Theorem 3 For the classical Jarník Besicovitch Theorem, such a localized theorem was obtained by Barral and Seuret [2], who were the first to consider localized Diophantine approximation Corollary 4 With the above notation and under the hypotheses of Theorem 3, we have + τ(v)u(v) lim ε 0 dim H([v ε, v + ε] E(f, y, τ)) + τ(v)l(v) We illustrate Theorem 3 and Corollary 4 by some examples If the family of functions f = (f n ) n in Theorem 3 is such that, for every x in I, the sequence (f n (x)) n increases sufficiently rapidly, then lim ε 0 dim H([v ε, v + ε] E(f, y, τ)) = + τ,

4 4 YANN BUGEAUD, LINGMIN LIAO, AND MICHA L RAMS independently of the family f This applies, for example, to the families of functions x n2, x n, 2 n x and x n The case f n (x) = a n x, where (a n ) n is an increasing sequence of positive integers, has been studied by Borosh and Fraenkel [4] (but only in the special case of a constant sequence y equal to 0) Let I be an open, non-empty, real interval They proved that dim H {x I : a n x < a τ n } = + s + τ, where s (usually called the convergence exponent of the sequence (a n ) n ) is the largest real number in [0, ] such that n a s ε n converges for any ε > 0 The case s = 0 of their result, which corresponds to rapidly growing sequences (a n ) n, follows from Theorem 3 The case a n = n for n corresponds to the Jarník Besicovitch Theorem We stress that the assumption (3) is satisfied only if (a n ) n increases sufficiently rapidly Questions of uniform Diophantine approximation were recently studied by Bugeaud and Liao [7] for the b-ary and β-expansions and by Kim and Liao [2] for the irrational rotations In this paper, we consider the uniform Diophantine approximation of the sequence ({x n }) n with x > For a real number B > and a sequence of real numbers y = (y n ) n in [0, ], set F (B, y) := {x > : for all large integer N, x n y n < B N has a solution n N} Our next theorem gives a lower bound for the Hausdorff dimension of F (B, y) intersected with a small interval Theorem 5 Let B > be a real number and y an arbitrary sequence of real numbers in [0, ] For any v >, we have ( ) log v log B 2 lim dim H([v ε, v + ε] F (B, y)) ε 0 log v + log B Unfortunately, we are unable to decide whether the inequality in Theorem 5 is an equality Observe that the lower bound we obtain is the same as the one established in [7] for a question of uniform Diophantine approximation related to b-ary and β-expansions Letting v tend to infinity, we have the following corollary Corollary 6 For an arbitrary sequence y of real numbers in [0, ] and any real number B >, the set F (B, y) has full Hausdorff dimension We end this paper with results on sequences ({x n }) n, with x >, which are badly distributed, in the sense that all of their points lie in a small interval As above, we take a more general point of view Consider a family of C strictly positive increasing functions f = (f n ) n defined on an open interval I R and such that f n (x), f n(x) > for all x I and for

5 DISTRIBUTION OF FRACTIONAL PARTS 5 all n Let δ = (δ n ) n be a sequence of positive real numbers such that δ n < /4 for n Set G(f, y, δ) := {x I : f n (x) y n δ n, n } We need the following hypotheses: (5) (6) ε > 0, n, x I, Our last main theorem is as follows inf x (v ε,v+ε) f n+ (x) sup x (v ε,v+ε) f n(x) δ n 2, log f n+ lim (x) log f n(x) = Theorem 7 Keep the above notation Under the hypotheses (), (5), and (6), for all v I, we have (7) lim dim H ([v ε, v + ε] G(f, y, δ)) = lim inf ε 0 log f n(v) + n j= log δ j log f n(v) log δ n We remark that our result extends a recent result of Baker [] In fact, in [], the author studied the special case f n (x) = x qn with (q n ) n being a strictly increasing sequence of real numbers such that lim (q n+ q n ) = + Our result also gives the following corollary Corollary 8 Let (a n ) n be a sequence of positive real numbers such that a n+ lim = + a n Then, for any sequence (y n ) n of real numbers, we have dim H {x R : lim a nx y n = 0} = n + 2 Basic tools We present two lemmas which serve as important tools for estimating the Hausdorff dimension of the sets studied in this paper Let [0, ] = E 0 E E 2 be a decreasing sequence of sets, with each E k a finite union of disjoint closed intervals The components of E k are called k-th level basic intervals Set F = k=0 E k We do not assume that each basic interval in E k contains the same number of next level basic intervals, nor that they are of the same length, nor that the gaps between two consecutive basic intervals are equal Instead, for x E k, we denote by m k (x) the number of k-th level basic intervals contained in the (k )-th level basic interval containing x, and by ε k (x) the minimal distance between two of them Set ε k (x) = min i k ε i(x) In the following, we generalize a lemma in Falconer s book [9, Example 46]

6 6 YANN BUGEAUD, LINGMIN LIAO, AND MICHA L RAMS Lemma 9 For any open interval I [0, ] intersecting F, we have dim H (I F ) inf lim inf x I F k log(m (x) m k (x)) log(m k (x)ε k (x)) Proof The proof is similar to that in the book of Falconer We define a probability measure µ on F by assigning the mass evenly Precisely, for k, let I k (x) be the k-th level interval containing x For x F and k, we put a mass (m (x) m k (x)) to the interval I k (x) Note that any two k-th basic intervals contained in the same (k )-th interval have the same measure One can check that the measure µ is well defined Now let us calculate the local dimension at the point x Let B(x, r) be the ball of radius r centered at x Suppose that ε k (x) 2r < ε k (x) The number of k-th level intervals intersecting B(x, r) is at most { } { } ( ) 2r min m k (x), ε k (x) + 4r 4r s min m k (x), m k (x) s, ε k (x) ε k (x) for any s [0, ] Thus ( ) 4r s µ(b(x, r)) m k (x) s (m (x) m k (x)) ε k (x) Hence log µ(b(x, r)) log r Let s be in (0, ) such that s < s log m k(x)ε k (x) s log(4r) + log(m (x) m k (x)) log r log(m (z) m k (z)) inf lim inf lim inf z I F k log m k (z)ε k (z) k log(m (x) m k (x)) log m k (x)ε k (x) Then s log m k (x)ε k (x) s log 4 + log(m (x) m k (x)) 0, for k large enough Therefore lim inf r 0 log µ(b(x, r)) log r s The proof is completed by applying the mass distribution principle (see [0], Proposition 23) We also have an upper bound for the dimension of the set I F Denote by I k (x) the length of the k-th basic interval I k (x) containing x Lemma 0 For any open interval I [0, ] intersecting F, we have dim H (I F ) sup lim inf x I F k log(m (x) m k (x)) log I k (x) Proof We define the same probability measure µ as in Lemma 9, ie, the interval I k (x) has measure (m (x) m k (x)) Then lim inf r 0 log µ(b(x, r)) log r lim inf k log µ(i k (x)) = lim inf log I k (x) k log(m (x) m k (x)) log I k (x) We finish the proof by applying again the mass distribution principle (see [0], Proposition 23)

7 DISTRIBUTION OF FRACTIONAL PARTS 7 3 Asymptotic approximation In this section, we prove Theorem 3 To see that Theorem is a special case of it, take the family of functions f defined by f n (x) = x n, n, we have u(v) = l(v) = and ( ) log b [v ε, v + ε] E f, y, log(v + ε) [v ε, v + ε] E Then, Theorem follows directly from Theorem 3 Now we prove Theorem 3 [v ε, v + ε] E(b, y) ( ) f, y, log b log(v ε) Proof of Theorem 3 Lower bound: We can assume that u(v) is finite, since otherwise there is nothing to prove Let us start by the simple observation about the condition () Given an integer n, set (3) { } log f η(n) = sup n (w) log f n(z) ; w, z [v ε, v + ε], f n(w) f n (z) Lemma If () and (3) hold, then lim η(n) = 0 Proof Assume this is not true Then there exists a sequence of integers (n i ) and a sequence of pairs of points (w i, z i ) such that f ni (w i ) f ni (z i ) and log f n i (w i ) log f n i (z i ) > Z > By compactness of [v ε, v + ε], taking a subsequence if necessary, we can assume that (w i ) i converges to some point w 0 By (3), f n(v) Hence, () gives us This implies that lim w i z i inf f n(x) = x [v ε,v+ε] inf x [v ε,v+ε] f n i (x) 0 as i, and hence any neighborhood of w 0 contains all except finitely many points w i, z i Thus, in any neighbourhood U of w 0 we have lim sup sup w,z U which is a contradiction with () log f n(w) log f n(z) > Z, Now we construct a nested Cantor set which is the intersection of unions of subintervals at level n i, where (n i ) i is an increasing sequence of positive integers which will be defined precisely later Suppose we have already well

8 8 YANN BUGEAUD, LINGMIN LIAO, AND MICHA L RAMS chosen this subsequence Let us describe the nested family of subintervals For each level i, we need to consider the set of points x such that f ni (x) y ni f ni (x) τ By the property f n (x) y n f n (x) τ, we take the intervals at level as I (k, v, f, y, τ) := [f n (k + y n f n (v + ε) τ ), f n (k + y n + f n (v + ε) τ )], with k being an integer in [f n (v ε) +, f n (v + ε) ] Suppose we have constructed the intervals at level i Let [c i, d i ] be an interval at such level A subinterval of [c i, d i ] at level i is such that [fn i (k + y ni f ni (d i ) τ ), fn i (k + y ni + f ni (d i ) τ )], with k being an integer in [f ni (c i ) +, f ni (d i ) ] By continuing this construction, we obtain intervals I i ( ) for all levels Finally, the intersection F of these nested intervals is obviously a subset of [v ε, v + ε] E(f, y, τ) Let z F and [c i (z), d i (z)] be the i-th level interval containing z Then we have (32) m i+ (z) f n i+ (w i ) (d i c i ) 2 f n i+ (w i ) 2f n i (d i ) τ where w i, z i [c i (z), d i (z)] Furthermore, (33) ε i+ (z) 2f n i+ (c i (z)) τ f n i+ (u i ) 2f n i+ (u i ), where u i [c i (z), d i (z)] Now we are going to define the subsequence (n i ) i f n i (z i ) 2, Lemma 2 Assume () and (2) For any γ > 0, we can find a subsequence (n i ) i such that (34) f n i+ (w) f n i+ (u) f n i (z) γ and for any small ε > 0, we have (35) and (36) x (v ε, v + ε), lim w, u [c i (z), d i (z)], log f n i (x) log f n i (x) = lim inf x (v ε,v+ε) f n i+ (x) sup x (v ε,v+ε) f n i (x) f ni (x) τ 2 log f n i (x) log f ni (x) =, If f n are linear then we do not need to assume (2), moreover we can choose (n i ) in such a way that we have (in addition to the other parts of the assertion) (37) lim log f ni (v) log f n i (v) = l(v)

9 DISTRIBUTION OF FRACTIONAL PARTS 9 Proof In the linear case (34) is automatically true, and to have (35) and (36) we just need that (n i ) i increases sufficiently fast (as will be clear from the proof for the general case) Hence, we will be free to choose (n i ) satisfying in addition (37) Let us proceed with the general case For any γ > 0, by Lemma, there exists n 0 N such that n n 0, η(n) < γ 2M, where M is the constant in assumption (2) Starting with this n 0, by the assumption (2), we can then construct a subsequence (n i ) i satisfying (38) γ 2η(n i ) M log f n (v) i+ log f n i (v) γ 2η(n i ) Observe that, as η(n i ) 0 by Lemma, the lefthand side of (38) implies the first part of (35) As u <, the second part of (35) follows The condition (36) will also follow, provided that n 0 was selected large enough We need now to prove (34) By (3), for any w, u in the interval [c i (z), d i (z)], (39) f n i+ (w) f n i+ (u) f n i+ (z)+η(ni) f n i+ (z) η(n i) = f n i+ (z) 2η(n i) Combining (38) and (39), we get (34) We continue the proof of the lower bound of Theorem 3 By (32) and (36), m i+ (z) f n i+ (w i ) 2f n i (d i ) τ f n 2 2, i (z i ) which then implies that F is non-empty Further, by (34), for any γ > 0, (30) By (32), (33) and (34), (3) m i+ (z) f n i+ (z) f n i (z) γ fn i (d i ) τ f n i (z i ) m i+ (z)ε i+ (z) f n i (z) γ fn i (d i ) τ 2f n i (z i ) Thus, (3) and (35) imply that log m i+ (z)ε i+ (z) is unbounded So by (30), (3) and (35) lim inf lim inf = lim inf log(m 2 (z) m i (z)) log m i+ (z)ε i+ (z) i j=2 (log f n j (z) γ log f n (z) τ log f j n j (d j ) log f n (z j j)) log 2 + log f n i (z i ) + γ log f n i (z) + τ log f ni (d i ) log f n i (z) log f n i (z i ) + γ log f n i (z) + τ log f ni (d i )

10 0 YANN BUGEAUD, LINGMIN LIAO, AND MICHA L RAMS Hence, by the definition of η(n i ), we have lim inf ( lim sup log(m 2 (z) m i (z)) log m i+ (z)ε i+ (z) + η(n i ) + γ + τ( + η(n i )) log fn i (d i) log f n i (d i ) In the linear case, log f ni (d i )/ log f n i (d i ) converges to l( lim d i ) In the general situation, we have lim sup log f ni (d i ) log f n u( lim i (d i ) d i) As γ can be chosen arbitrarily small, η(n i ) 0 by Lemma, and lim d i [v ε, v + ε], the lower bound is obtained by applying Lemma 9 Upper bound: Since for all x [v ε, v + ε] E(f, y, τ), we have f n (x) y n < f n (x) τ for infinitely many n Then the set [v ε, v + ε] E(f, y, τ) is covered by the union of the family of intervals I n (k) := [fn (k + y n f n (v ε) τ ), fn (k + y n + f n (v ε) τ )], where k [f n (v ε), f n (v + ε)] is an integer Note that the length of the interval I n (k) satisfies I n (k) 2f n(v ε) τ f n(z) The number of the intervals at level n is less than f n (v + ε) f n (v ε) 2εf n(w) Thus for s > 0 (32) n= k [f n(v ε),f n(v+ε)] I n (k) s ) for some z (v ε, v + ε) for some w (v ε, v + ε) n= ( 2εf n(w) 2fn (v ε) τ ) s f n(z) By the definition of l(v), for any η > 0, there exists n 0 = n 0 (η) N such that for any n n 0 f n (v ε) > f n(v ε) l(v ε) η Thus by ignoring the first n 0 terms, we have (32) is bounded by 2 +s ε f n(w) f n(z) s (f n(v ε) ) τs(l(v ε) η) (33) n=n 0 Hence by the assumption (3) if s > lim sup log f n(w) log f n(z) + τ(l(v ε) η) log f n(v ε)

11 the sum in (32) converges By (), Therefore DISTRIBUTION OF FRACTIONAL PARTS log f lim n(w) log f n(z) =, lim lim dim H[v ε, v + ε] E(f, y, τ) ε 0 log f n(w) log f n(v ε) = + τl(v) 4 Uniform Diophantine approximation In this section, we study the uniform Diophantine approximation of the sequence ({x n }) n with x > Recall that for any sequence of real numbers y = (y n ) n in [0, ], we are interested in the set F (B, y) := {x > : for all large integer N, x n y n < B N has a solution n N} For any v F (B, y), for any ε > 0, we will give a lower bound for the Hausdorff dimension of [v ε, v+ε] F (B, y) To this end, we investigate the uniform Diophantine approximation and asymptotic Diophantine approximation together We consider the following subset of [v ε, v + ε] F (B, y) F (v, ε, b, B, y) := {z [v ε, v + ε] : z n y n < b n for infinitely many n and N, z n y n < B N has a solution n N} The proof of Theorem 5 will be completed by maximizing the lower bounds of F (v, ε, b, B, y) with respect to b > B Proof of Theorem 5 We first construct a subset F F (v, ε, b, B, y) Suppose that b = B θ with θ > Let n k = θ k Consider the points z such that z n k y nk < b n k Then one can check that z F (v, ε, b, B, y) = F (v, ε, B θ, B, y) = F (v, ε, b, b θ, y) We do the same construction as in Section 3 We will obtain a Cantor set F F (v, ε, b, b θ, y), which is the intersection of a nested family of intervals with m k (z) = 2n k+c k (z) n k+ n k b n k dk (z) n k and ) ( 2 ε k (z) = b n k+ n k+ d k (z) n k+, where [c k (z), d k (z)] is the k-th level interval containing z By the choice of n k, we will have the following estimations: ( ) ck (z) θ k θ b θk d k (z) m k (z) 2(θk+ )c k (z) θk+ θ k b θk d k (z) θk c k (z) θk (θ )

12 2 YANN BUGEAUD, LINGMIN LIAO, AND MICHA L RAMS and Since ε k (z) 2θ k+ d k(z) θk+ d k (z) c k (z) b n k n k c k (z) n k b θk is much more smaller than /θ k, ( ) ck (z) θ k ( = d k(z) c k (z) d k (z) d k (z) Then m k (z) θ ( 2 ck (z) θ ) θ k b and m k (z)ε k (z) ( 4θ k ck (z) θ ) θ k b d k (z) θ Thus by Lemma 9, for any z F, we have lim inf k lim inf k = lim inf k = θ 2 θ+ ) θ k ( z θ b 2 ) θ k ( ) θ k 2 θ+3 θ k bz log(m (z) m k (z)) log m k (z)ε k (z) ( ) (θ ) log z log b k j= θj θ k log bz (θ ) log z log b (θ ) log bz (θ ) log z log b (θ ) log bz θk θ k Hence, by the relation b = B θ, we deduce that the Hausdorff dimension of the set F (v, ε, b, B, y) = F (v, ε, B θ, B, y) is at least equal to (θ ) log(v ε) log b (θ ) log(b(v ε)) = log(v ε) θ θ log B log(v ε) + θ log B Taking θ in the left side of the equality, we get the lower bound log(v ε)/ log(b(v ε)) for the Hausdorff dimension of the set considered in Theorem : [v ε, v + ε] E(b, y) ={v ε x v + ε : x n y n < b n for infinitely many n} By maximizing the right side of the equality with respect to θ >, we obtain the lower bound ( ) log(v ε) log B 2 log(v ε) + log B for the Hausdorff dimension of the set [v ε, v + ε] F (B, y) ={v ε x v + ε : N, x n y < B N has a solution n N} By letting ε tend to 0, this completes the proof of Theorem 5,

13 DISTRIBUTION OF FRACTIONAL PARTS 3 5 Bad approximation In this section, we study the bad approximation properties of the sequence ({x n }) n, where x > Let q = (q n ) n be a sequence of positive real numbers and y = (y n ) n be an arbitrary sequence of real numbers in [0, ] Define and, for v >, define G(q, y) = {x > : lim xqn y n = 0}, G(v, q, y) = { < x < v : lim xqn y n = 0} Recently Baker [] showed that if q = (q n ) n is strictly increasing and lim (q n+ q n ) =, then the set G(q, y) has Hausdorff dimension We want to generalize Baker s result Consider a family of C functions f = (f n ) n from an interval I R to R such that f n(x) for all x I and for all n Let δ = (δ n ) n be a sequence of positive real numbers tending to 0 For ε > 0, set G(ε, v, f, y, δ) := {v ε < x < v + ε : f n (x) y n δ n, n } To prove Theorem 7, we need to estimate dim H G(ε, v, f, y, δ) Sketch proof of Theorem 7 Lower bound: We do the same construction as in the proof of the lower bound in Theorem 3 If the right-hand side inequality in (38) is satisfied, that is, if (5) log f n+ (v) log f n(v) γ 2η(n), for some γ > 0, for large enough n, and for η defined in (3), then the distortion estimation (34) holds and we estimate the dimension in exactly the same way as in Theorem 3 If, however, (5) is not satisfied, that is, at some place f n is too sparse, with log f n+ (v) log f n(v) then we can apply the idea of Baker ([], page 69): we add some new functions f m between f n and f n+, in such a way that the resulting, expanded, sequence of their logarithms of derivatives is not too sparse anymore We also add some δ m = for each added f m Observe that the right-hand side of (7) does not change Naturally, the resulting set G(ε, v, f, y, δ) is exactly the same as G(ε, v, f, y, δ) So, for the lower bound, we need only to estimate the lower bound of dim H G(ε, v, f, y, δ) This means that we can freely assume that (5) holds We will construct a subset of G(ε, v, f, y, δ) which is the intersection of a nested family of subintervals I n ( ) For n =, by the property f (x) y δ, we take the intervals at level as I (k, v, f, y, δ) := [f (k + y δ ), f (k + y + δ )], with k being an integer in [f (v ε) +, f (v + ε) ]

14 4 YANN BUGEAUD, LINGMIN LIAO, AND MICHA L RAMS Suppose we have constructed the intervals at level n Let [c n, d n ] be an interval at this level A subinterval of [c n, d n ] at level n is [fn (k + y n δ n ), fn (k + y n + δ n )], with k being an integer in [f n (c n ) +, f n (d n ) ] By continuing this construction, we obtain intervals I n ( ) for all levels Finally, the intersection F of these nested intervals is obviously a subset of G(ε, v, f, y, δ) Let z F and [c n (z), d n (z)] be the n-th level interval containing z Then by (5) and m n+ (z) f n+(w n ) (d n c n ) 2 f n+(w n ) ε n+ (z) 2δ n+ f n+ (u n) 2f n+ (u n) 2δ n f n(z n ) 2 2, with w n, z n, u n [c n (z), d n (z)] As we are assuming (5), we have (34) and then for any γ > 0 and Thus, By (6), we have m n+ (z) f n+(z) f n(z) γ m n+ (z)ε n+ (z) f n(z) γ log(m 2 (z) m n (z)) log m n+ (z)ε n+ (z) log f n(z) log f n (z) + δ n f n(z n ) δ n 2f n(z n ) log δ j + n log f j (z) γ f j j= j= (z j) log 2 + log f n(z n ) + γ log f n(z) log δ n lim inf lim inf log(m 2 (z) m n (z)) log m n+ (z)ε n+ (z) log f n(z) + n j= log δ j log f n(z n ) + γ log f n(z) log δ n Since γ can be chosen arbitrary small and z n tends to z, by () we have lim inf log(m 2 (z) m n (z)) lim inf log m n+ (z)ε n+ (z) log f n(z) + n j= log δ j log f n(z) log δ n Hence the lower bound of Theorem 7 is obtained by Lemma 9 Upper bound: We will apply Lemma 0 For each basic interval I n (z), by (), we have for any γ, for n large enough δ n f n(z)f n I n(z) δ nf n (z)γ (z)γ f n(z)

15 DISTRIBUTION OF FRACTIONAL PARTS 5 Thus, Hence, m n (z) I n (z) f n(z)f n (z) γ δ n f n 2 (z)γ f n (z) f n(z)f n (z) γ Therefore, by (6), n j=2 n m j (z) f n(z) δ j j= n j= f j (z)2γ f (z) lim inf log(m (z) m n (z)) log I n (z) By Lemma 0, we conclude the proof lim inf log f n(z) + n j= log δ j log f n(z) log δ n References [] S Baker, On the distribution of powers of real numbers modulo, Uniform Distribution Theory 0, no2, (205), [2] J Barral, S Seuret, A localized Jarnik-Besicovich theorem, Adv Math 226 (4) (20), [3] V Beresnevich and S Velani A Mass Transference Principle and the Duffin Schaeffer conjecture for Hausdorff measures, Ann of Math (2) 64 (3) (2006), [4] I Borosh and A S Fraenkel A generalization of Jarník s theorem on Diophantine approximations, Indag Math 34 (972), [5] Y Bugeaud, Distribution modulo one and Diophantine approximation Cambridge Tracts in Mathematics 93, Cambridge, 202 [6] Y Bugeaud, A Dubickas, On a problem of Mahler and Szekeres on approximation by roots of integers, Michigan Math J 56 (2008), [7] Y Bugeaud and L Liao, Uniform Diophantine approximation related to b-ary and β-expansions, Ergod Th & Dynam Sys, 36, no, (206), 22 [8] Y Bugeaud, N Moshchevitin, On fractional parts of powers of real numbers close to Math Z 27 (202), no 3-4, [9] K J Falconer, Fractal Geometry, Mathematical Foundations and Application, Wiley, 990 [0] K J Falconer, Techniques in Fractal Geometry, Wiley, 997 [] J P Kahane, Sur la répartition des puissances modulo, C R Math Acad Sci Paris 352 (204), no 5, [2] D H Kim and L Liao, Dirichlet uniformly well-approximated numbers, Int Math Res Not IMRN To appear [3] J F Koksma, Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins, Compositio Math 2 (935), [4] J F Koksma, Sur la théorie métrique des approximations diophantiques, Indag Math, 7 (945), [5] K Mahler and G Szekeres, On the approximation of real numbers by roots of integers, Acta Arith, 2 (967), [6] AD Pollington, The Hausdorff dimension of certain sets related to sequences which are not dense mod, Quart J Math Oxford Ser (2) 3 (980), 35 36

16 6 YANN BUGEAUD, LINGMIN LIAO, AND MICHA L RAMS Yann Bugeaud, IRMA UMR 750, CNRS, Université de Strasbourg, 7, rue René Descartes, Strasbourg, France address: bugeaud@mathunistrafr Lingmin Liao, LAMA UMR 8050, CNRS, Université Paris-Est Créteil, 6 Avenue du Général de Gaulle, 9400 Créteil Cedex, France address: lingminliao@u-pecfr Micha l Rams, Institute of Mathematics, Polish Academy of Sciences, ul Śniadeckich 8, Warszawa, Poland address: MRams@impanpl