CENTRAL LIMIT THEOREM AND DIOPHANTINE APPROXIMATIONS. Sergey G. Bobkov. December 24, 2016

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1 CENTRAL LIMIT THEOREM AND DIOPHANTINE APPROXIMATIONS Sergey G. Bobkov December 24, 206 Abstract Let F n denote the distribution function of the normalized sum Z n = X + +X n /σ n of i.i.d. random variables with finite fourth absolute moment. In this paper, polynomial rates of convergence of F n to the normal law with respect to the Kolmogorov distance, as well as polynomial approimations of F n by the Edgeworth corrections modulo logarithmically growing factors in n are given in terms of the characteristic function of X. Particular cases of the problem are discussed in connection with Diophantine approimations. Introduction Let X, X, X 2,... be independent, identically distributed random variables with mean zero, variance σ 2 σ > 0 and finite 3-rd absolute moment β 3 = E X 3. Denote by F = P{X } the distribution function and by ft = E e itx the characteristic function of X. The Berry-Esseen theorem provides a standard rate of approimation of the distribution functions F n = P{Z n } of the normalized sums Z n = X + + X n σ n by the standard normal distribution function Φ with density ϕ = 2π e 2 /2 R. Namely, up to a numerical constant c, we have β 3 F n Φ c σ 3 n. In general, higher order moment assumptions do not improve this rate, as can been seen on the eample of lattice distributions F. Nevertheless, under the Cramér condition lim t ft <,. Key words: Central limit theorem, Diophantine approimation, Edgeworth epansions. MSC 60F Address: School of Mathematics, University of Minnesota, 27 Vincent Hall, 206 Church St. S.E., Minneapolis, MN USA. bobkov@math.umn.edu

2 2 it is possible to slightly correct the limit law by allowing dependence in n, so as to improve the rate of approimation. In particular, consider an Edgeworth correction of the 3-rd order Φ 3 = Φ α 3 6σ 3 n 2 ϕ, α 3 = EX 3,.2 which also depends on n, ecept for the case α 3 = 0 when Φ 3 = Φ. It is well-known that, if the 4-th absolute moment β 4 = EX 4 is finite, the uniform deviations n = F n Φ 3 are at most of order /n. Moreover, with higher order moments assumptions, the corresponding higher order Edgeworth corrections called also Edgeworth epansions provide an error of approimation decaying as powers of / n, cf. e.g. P]. Without the Cramér condition., the problem of possible rates is rather delicate, as the order of magnitude of n depends on arithmetical properties of the point spectrum of F n. This was already emphasized by Esseen, who established the following general result cf. E], pp : If X has a non-lattice distribution equivalently, ft < for all t > 0, and if the 3-rd absolute moment of X is finite, then n = o n as n..3 It seems that not much has been said in literature in addition to this theorem see, however, a cycle of papers Ch]. The aim of these notes is to refine.3 by connecting possible polynomial rates for n with behavior of the characteristic function ft at infinity. Let us stress that, although the lack of the Cramér property forces F not to have an absolutely continuous component, the class of probability distributions with lim t ft = is etremely rich and interesting including discrete and many purely singular continuous probability measures. For simplicity, we focus on intermediate rates between n and n for n. Let us state the relationship, by using the notation Õtp for the growth rate Ot p log t q with some q, and similarly Õn p for On p log n q. Theorem.. Suppose that β 4 <. Given p 2, the following two properties are equivalent: = ft Õtp as t ;.4 n = Õ n 2 p as n..5 A more precise formulation reflecting appearance of the logarithmically growing factors in Õ in.4-.5 will be given in Sections 3 and 5. As for the restriction p 2, it may actually be relaed to p > 0 under higher moment assumptions by adding to Φ 3 other terms in the corresponding Edgeworth epansions.

3 3 Let us illustrate Theorem. in a simple discrete situation. As is standard, we denote by the distance from a real number to the closest integer. Given an irrational real number α, define the quantity ηα = { } η > 0 : lim inf n nη nα = 0 = inf { } η > 0 : inf n nη nα > 0. One says that α is of type η = ηα and calls +η an irrationality eponent of α. Equivalently, the value of η is an optimal one, for which the Diophantine inequality α p < q q +η ε has infinitely many rational solutions p q with any fied ε > 0 cf. e.g. K-N], B-B-S]. Thus, this quantity provides an important information on how well the number α may be approimated by rationals. By Dirichlet s theorem, necessarily η, and actually η s fill the half-ais, ] including the case η = which describes the class of Liouville s numbers. Applying Theorem. with p = 2η, one may derive the net characterization. Corollary.2. Given an irrational number α, pose that the random variable X takes the values ± and ±α each with probability /4. Then, α is of finite type η if and only if, for any ε > 0, F n Φ = O n 2 +ε 2η as n..6 A similar description continuous to hold when X takes the values ± ± α. In this case, one may write X = X + αx in the sense of laws, where X and X are independent random variables with a symmetric Bernoulli distribution on {, }. While for X and αx separately, the corresponding deviations n are of order / n, we see that the convolution structure in the underlying distribution F may essentially improve the rate. For eample, by Roth s theorem cf. C], S-2], we have η = for any irrational algebraic α, and then.6 becomes n = On +ε. If α is a quadratic irrationality, or more generally, a badly approimable number, one may sharpen the rate to n = O n log n. Although in these eamples, such α s form a set of Lebesgue measure zero, a slightly worse rate n = O n log n 3 +ε 2 can be derived for almost all values of α on the line see Section 7 for details. It is interesting to compare relation.6 with a statement about an asymptotic behavior of empirical measures F n = n δ n {kα}, k= where {} stands for the fractional part and δ denotes a point mass at a given point one may similarly consider the sequence kα and use the identity = min{{}, {}}. By Weyl s criterion, F n are weakly convergent to the uniform distribution on 0,, as long as α is

4 4 irrational. Results by Hecke, Ostrowski and Behnke in 920 s quantify this convergence: For any ε > 0, with some positive c 0 = c 0 α, ε and c = c α, ε, we have c 0 n η ε Fn c n η +ε,.7 0<< where η = ηα K-N]. Although there is some difference between.6 and.7, the two rates turn out to be in essence the same in the critical case η =. Let us also mention that, for quadratic irrationalities α, an asymptotic behavior of F n has been comprehensively studied in the recent times by Beck, cf. B]. The paper is organized as follows. In section 2 we remind a basic Berry-Esseen-type bound for the distributions F n which is applicable to reach the rate of approimation of F n by Φ 3 potentially up to order /n. Here we also eplain the sufficiency part in Theorem.. In Sections 3-4 we discuss non-uniform bounds on F n Φ 3 together with bounds on the difference between the Fourier-Stieltjes transforms of F n and Φ 3. The necessity part in Theorem. is considered separately in Section 5. Section 6 deals with Diophantine inequalities, where Corollary.2 is derived, actually in a somewhat more general and precise form. Applications of this corollary are clarified in Section 7. 2 Berry-Esseen inequality. Sufficiency part in Theorem. The derivation of uniform estimates on the difference between distribution functions, say F and G, is commonly based on a general Berry-Esseen bound F G involving the Fourier-Stieltjes transforms ft = T e it df, gt = 0 ft gt t dt + D T e it dg T > 0, 2. t R. In fact, in 2., G may be an arbitrary differentiable function of bounded variation on the real line such that G = 0, G =, and G D E], P2]. With this approach, the implication.4.5 is rather standard although we cannot give an eact reference. For completeness, we remind the basic argument in the special situation as in Theorem. which yields a bound on n = F n Φ 3. Namely, one may apply 2. with F n in place of F and with G = Φ 3. The Fourier-Stieltjes transform of F n is just the characteristic function of Z n given by f n t = f t σ n n, where f is the characteristic function of X. The Fourier-Stieltjes transform of Φ 3 is g 3 t = e t2 /2 + α 3 6σ 3 n it3 e t2 / We will denote by c a positive absolute constant which may be different in different places.

5 5 Lemma 2.. Suppose that β 4 is finite. For all n and T c n β 4 σ 4 n + T T σ n + σ β4 σ β4, ft n dt. 2.3 t Proof. Put T 0 = σ2 β4 n and introduce the Lyapunov coefficients Ls = βs σ n s 2 s 2 β s = E X s, which we need for s = 3 and s = 4. Since the function s Ls /s 2 is non-decreasing in s > 2, we have L 3 L /2 4 and thus α 3 σ 3 n β 3 σ 3 n = L 3 L /2 4 = T 0. Hence, according to definition.2, Φ 3 c + T 0 for 0 and Φ 3 c + T 0 for 0. This implies that 2.3 holds automatically in case T 0 for a suitable c. Thus, we may assume that T 0, i.e., n β 4 /σ 4. Now, let us look at the integrand in 2.. It is known that f n t is approimated by g 3 t on the interval t /L 3 with an error of order /n using Taylor s epansion for ft near zero and the product structure of f n t. In particular, for a smaller interval t T 0, there is a well-known estimate f n t g 3 t c β 4 σ 4 n min{, t4 } e t2 /8. Therefore, with T in place of T, inequality 2. yields, for any T T 0, c n β 4 σ 4 n + T f n t g 3 t + dt. 2.4 T T 0 t According to 2.2 and using the assumption T 0, we also have which implies g 3 t + 6 t3 e t2 /2 <.3 e t2 /8 t 0, 2.5 T T 0 g 3 t t dt c T 0 e t2 /8 dt < 2c e T 2 0 /8 < 6 c T 2 0 = 6c β 4 σ 4 n. As a result, 2.4 is simplified to c n β 4 σ 4 n + T f n t + dt. T T 0 t Putting T = T σ n and changing the variable, we arrive at 2.3. equivalent to T σ β4 Note that T T 0 is Using Lemma 2., one obtains the statement of Theorem. in one direction.

6 6 Proposition 2.2. Suppose that β 4 is finite and let, for some p > 0 and q R, ft = O t p log t q as t. Then n = O n 2 p log n q+ p + n. 2.6 For p < 2 with arbitrary q and for p = 2 with q, relation 2.6 reduces to n = O n, while in other cases, n = O n 2 p log n q+ p. In particular, the hypothesis ft = Õtp with p 2 implies n = Õn 2 p. Proof. Suppose that q 0. By the assumption, and since necessarily X has a non-lattice distribution, we have for all T t 0 = σ β4, MT = ma ft a t 0 t T T p log q 2 + T with some constant a > 0. Using u e u, we then get { ft n MT n ep na T p log q 2 + T }, so that Thus, by 2.3, T t 0 ft n t { na } dt ep T p log q logt/t T c n β 4 σ 4 n + { T σ n + ep na } T p log q logt/t T Let us take T = T n = bn /p log n r with parameters r 0, b > 0 to be precised later on and assuming that n is large enough. Then T p n bn log n rp, log2 + T n p log n + Olog log n, and This gives log q 2 + T n p q log nq + O log n q log log n. Tn p log q 2 + T n b p q n log nq rp + O n log n q rp log log n.

7 7 Choosing r = q + /p, the above is simplified to Tn p log p 2 + T n b p q n log n + O log n log log n, and then na Tn p log p 2 + T n apq b log n + Olog log n 2 log n, where the last inequality holds true with b = ap q /3 for all n large enough. In this case, the last term in 2.7 is estimated from above by O/n. In case q = 0 with choice r = /p, we clearly arrive at the same conclusion. Therefore, 2.7 yields n = O n + = O T n n n + n p 2 log n r, r = q + p. 3 Non-uniform bounds based on uniform bounds Suppose that a given distribution function F is well approimated by some function of bounded variation G such that G = 0, G =, in the sense of the Kolmogorov distance = F G. Based on this quantity, one would also like to see that F G decays polynomially fast for growing. To this aim one may use moment assumptions together with some possible properties of G related to its behavior at infinity. Lemma 3.. Suppose that F and G have finite and equal second moments: Then, for any a > 0, ] 2 F G 2 df = 4a ma a { a 2 dg dg 2 G ], a 2 G ]}. 3.2 Proof. For a, we have 2 F G a 2 which is dominated by the right-hand side of 3.2. So, when estimating 2 F G, one may assume that > a and that ±a are the points of continuity of both F and G. Integrating by parts, we have a a y 2 df y = a 2 F a Ga a 2 F a G a 2 a a y F y Gy dy + a a y 2 dgy.

8 8 Hence a a a y 2 df y 4a 2 + y 2 dgy a which implies, by the moment assumption 3., y 2 df y 4a 2 + so, y a On the other hand, in case a, y 2 df y y 2 df y Since also we get y a 2 G F y a y 2 dgy F = 2 G F + 2 G, y a y 2 df y + 2 G ]. a 2 F G 2 G 2 G ], a 2 F G By a similar argument, if a, 2 F G Therefore, in both cases, 2 F G It remains to involve 3.3. y a y a y a y 2 df y + 2 G ]. a y 2 df y + 2 G ]. a { y 2 df y + ma 2 G ], 2 G ]}. a a In particular, if G as measure is ported on the interval a, a], then, under the moment assumption 3., we have ] 2 F G 4a In the general non-compact case, in order to optimize inequality 3.2 over the variable a, an etra information is needed about the behavior of G. For eample, let us require that, for some parameters A, B > 0, G Ae 2 /B for 0, G Ae 2 /B for The function te t is decreasing for t. Hence, if a B, then 2 G A 2 e 2 /B Aa 2 e a2 /B.

9 9 In addition, a 2 dg = a 2 Ga + 2 Aa 2 e a2 /B + 2A a a G d e 2 /B d = A a 2 + B e a2 /B 2Aa 2 e a2 /B. Similar bounds also hold in the region a. Hence, inequality 3.2 yields 2 F G 4a 2 + 5Aa 2 e a2 /B, a B. Moreover, choosing a 2 = B loge +, the above right-hand side becomes 4B log e + + 5AB e + log e + 4B + 5AB log e +. Note that the parameters A and B may not be arbitrary. Applying the hypothesis 3.5 at the origin = 0, we get G0 + G0 2A. So, necessarily A 2 and hence 4 + 5A 3A. Thus, applying Lemma 3., we arrive at the following assertion. Proposition 3.2. Under the assumptions 3. and 3.5, ] 2 F G 3 AB log e In case of the normal distribution function G = Φ, we have Φ 2 e 2 /2 0, so, the conditions 3. and 3.5 are fulfilled with A = 2 and B = 2. Hence ] 2 F Φ 3 loge + /, 3.7 provided that 2 df =. In a slightly different form, this inequality can be found in P2], Ch. V, Theorem, pp where it is attributed to Kolodyazhnyi K]. The proof of a more general Lemma 3. given above follows the same line of arguments as in P2]. As for Proposition 3.2, we will need it in case G = Φ 3. 4 Deviations of characteristic functions The non-uniform bound 3.6 allows one to control deviations of the Fourier-Stieltjes transform f of the distribution function F from the Fourier-Stieltjes transform of G. Recall that G is assumed to be a function of bounded variation such that G = 0 and G =. From 3.6 it follows that, for any b > 0, ] b F G b AB log e +,

10 0 and therefore W F, G π b F G d b AB log e + ] 3 AB = π b + b log e + ]. Optimizing the right-hand side over all b > 0 and using π 26 < 6.02, we arrive at In particular, we get: W F, G 6.02 AB log /2 e + Proposition 4.. Under the assumptions 3. and 3.5, for all t R, where = F G. ft gt 6.02 AB t log /2 e + This bound follows from 4. via the the identity ft gt = it e it F G d.. 4., 4.2 The logarithmic term in 4.2 may be removed for compactly ported distributions G, even if F is not compactly ported. Indeed, starting from 3.4, for any b > 0, ] b F G b 2 + 4a 2, and therefore W F, G π b b2 + 4a 2 = π b + 4a2 ] = 4πa, b where in the last equality we take an optimal value b = 2a. Hence, if G is ported on the interval a, a] as measure and has the same second moment as F, then ft gt 4πa t t R. Now, let us return to the setting of Theorem. and specialize Proposition 4. to As was eplained in Section 2, G = Φ 3 = Φ α 3 6σ 3 n 2 ϕ. α 3 σ 3 n as long as n β 4/σ 4. In this case, for any 0, Φ 3 Φ + α 3 6σ 3 n 2 ϕ 2 e 2 / π 2 e 2 /2.

11 Being multiplied by e 2 /4, the above right-hand side attains maimum at zero, hence Φ π e 2 /4 < 0.57 e 2 /4. Thus, the assumption 3.5 is fulfilled with A = 0.57 and B = 4. We then get: Corollary 4.2. Suppose that β 4 is finite. For all n β 4 /σ 4, the characteristic function f n t of Z n satisfies, for all t R, where n = F n Φ 3. f n t g 3 t 24.2 t n log /2 e + n, 4.3 In fact, when α 3 = 0, we have Φ 3 = Φ, and the requirement n β 4 /σ 4 together with the 4-th moment assumption are not needed in Corollary 4.2. Moreover, since AB = for G = Φ in 3.5, from 3.7 we obtain a better numerical constant. Namely, f n t e t2 / t n log /2 e + n. 5 Necessity part in Theorem. Keeping the setting of Theorem., one may use the deviation inequality 4.3 to show that ft is properly bounded away from and thus to reverse the statement of Proposition 2.2. In this direction, only the finiteness of the 3-rd absolute moments is needed which is necessary, since α 3 participates in the definition of Φ 3. Proposition 5.. Suppose that, for some p > 0 and q R, n = O n 2 + p log n q as n. Then ft = O t p log t p +q 2 as t. 5. Proof. By the assumption, n log /2 e + = O n 2 p log n q+ 2. n Hence, using the upper bound 2.5 on g 3 t, 4.3 yields, for all n β 4 /σ 4, f n t.3 e t2 /8 + c t n 2 p log n q+ 2.

12 2 Here in the region t n, the second term on the right-hand side dominates the first one. Replacing t with t n, we therefore obtain that ft/σ n c p,q t n /p logn + q+/2, t, 5.2 with some p, q-dependent constant c p,q. Assuming that t e, let us choose with parameters A and r > 0. In this case, and n = 2At p log t r ] 5.3 n /p At p log t r /p = A /p t log t r/p logn + log4at p log t r = log4a + p log t + r log log t < log4a + p + r log t < p + r + log t, where in the last inequality we require that t 4A. Hence n /p logn + q+/2 A /p t log t r/p p + r + q+/2 log t q+/2 = A /p c p,qt, where we chose r = pq + /2 on the last step. Hence, with some p, q-dependent constant, 5.2 is simplified to ft/σ n c p,q A /p, which can be made smaller than /e by choosing a sufficiently large value of A. Thus, recalling 5.3, we have ft/σ e /n 2n 4At p log t r, which yields Diophantine inequalities Turning to Corollary.2 and other applications of Theorem., it makes sense to describe a somewhat more general situation. First let us list a few simple metric properties of the function in real variable. This function is even, -periodic, and satisfies, for all real, y, i ; ii + y + y ; iii y y. In addition, cosπ ep{ π 2 2 /2}, 4 2 cosπ π

13 3 Inequalities in 6. are elementary, and we omit the proofs. Below, we denote by n the closest integer to, so that = n for definiteness, let n = n in case = n + /2. Lemma 6.. Given real numbers α,..., α m, pose that ma k m nα k εn > 0 for all integers n. Then, for all t real, where c = + ma k m α k. t 2 + tα tα m 2 c 2 εnt 2, 6.2 Proof. One may assume that all α k > 0. Let t = n + γ, γ = t, with n = nt. If t cεn, c > 0, then automatically Mt ma { t, tα,..., tα m } cεn. Now, pose that t < cεn. By the assumption, nα k εn for some k m. Since tα k = nα k + γα k, we get, applying properties i and iii: tα k nα k γα k nα k γα k = nα k t α k cα k εn. Here cα k = c for c = +α k, and then tα k cεn in both cases. Hence, Mt εn + α k. Clearly, 6.2 with integer values t = n returns us to the assumption up to the α k -depending factor. Let us now consider a system of m Diophantine inequalities α k r k < εn, k =,..., m n, n n about which one is usually concerned whether or not it has infinitely many integer solutions r,..., r m, n. Here, we choose the particular functions εn = c n η logn + η and consider the opposite property: lim inf n n η log n η One may rephrase this in terms of the characteristic function ] ma{ nα,..., nα m } > ft = cost cosα t... cosα m t 6.4 of the sum X = ξ 0 + α ξ + + α m ξ m, where ξ k are independent Bernoulli random variables, taking the values ± with probability /2. Lemma 6.2. Given α,..., α m R and η > 0, η R, relation 6.3 is equivalent to the property that the characteristic function f in 6.4 satisfies ft = O t 2η log t 2η as t. 6.5

14 4 Proof. For 6.3 to hold, it is necessary that at least one of α k be irrational. Moreover, this relation may be strengthened to ma nα k k m c n η logn + η, n, 6.6 with some constant c > 0 independent of n. Moreover, according to Lemma 6. with εn as above, we see that 6.6 is equivalent to t 2 + tα tα m 2 c t 2η log t 2η, t 2 real 6.7 modulo positive constants. Combining 6.7 with the first inequality in 6. yields { fπt ep π2 t 2 + α t tα m 2} { c } ep 2 t 2η, log t 2η which thus leads to the required relation 6.5. Conversely, 6.5 yields fπt so that for the integer values t = n we get c t η logt + η, t, 6.8 c n 2η logn + 2η fπn = δ... δ m, δ k = cosπnα k. Since the right-hand side is greater than or equal to δ + + δ m, we obtain c n 2η logn + 2η δ + + δ m. Recalling 6., we have δ k π2 2 nα k 2 and thus c n 2η logn + 2η This gives 6.6 and therefore 6.3. π2 2 m k= nα k 2 mπ2 2 ma nα k 2. k m A similar conclusion continues to hold for other characteristic functions including m ft = p 0 cost + p k cosα k t, 6.9 where p k are fied positive parameters such that p p m =. Indeed, by 6., fπt p 0 4 t 2 + k= m p k 4 αk t 2 k= p t 2 + α t α m t 2, p = 4 min 0 k m p k.

15 5 Starting from , we would obtain again 6.5. Conversely, 6.5 leads to 6.8, which at the even integer values t = 2n yields c m m n 2η f2πn = p 0 + p k cos2πnα k = 2 p k δ 2 logn + 2η k, where now δ k = sinπnα k. Using sinπ π, the above inequality yields c m n 2η 2π 2 p k nα k 2 2π 2 ma logn + nα k 2. 2η k m As a result, we arrive at: k= Lemma 6.3. The assertion of Lemma 6.2 is also true for all characteristic functions f of the form 6.9. We are prepared to prove Corollary.2, in fact in a more precise and general form, if we apply Propositions 2.2 and 5.. Let us return to the setting of Theorem. in which we will assume that the random variable X has a characteristic function f given by 6.4 or 6.9. Equivalently, if we denote by B α = 2 δ α + 2 δ α the symmetric Bernoulli measure ported on { α, α}, the distribution F of X may be written as measure in either of the two forms m F = B B α B αm, F = p 0 B + p k B αk p k > 0, p p m =. Since any such measure is symmetric about the origin, the uniform distance in Theorem. is defined by n = F n Φ. Then k= Proposition 6.4. Given α,..., α m R, pose that with some η, η R, lim inf n η log n η ma { nα,..., nα m }] > n k= k= n = O n 2 2η log n η 6. with η = 2η + 2η in case η > and η = ma { 2η + 2, 0 } in case η =. Conversely, if 6. holds with some η > 0, η R, then 6.0 is fulfilled with η = η 2 +η. Indeed, starting from the hypothesis 6.0, we obtain 6.5, so that the condition of Proposition 2.2 is fulfilled with p = 2η and q = 2η. Hence, by Proposition 2.2, n = O n 2 p log n q+ p + n, i.e. 6.. Conversely, 6. ensures that the condition of Proposition 5. is fulfilled with p = 2η and q = η. Therefore, ft = O t p log t p +q 2 = O t 2η log t 2η 2 +η, which is 6.5 with 2η = 2η 2 + η.

16 6 7 Special values of α and typical behavior of n Let us restrict the setting of Proposition 6.4 to the case m = and assume that the distribution F of X has a convolution structure, i.e., X = X +αx, where X, X are independent random variables with a symmetric Bernoulli distribution on {, }. The corresponding characteristic function is then given by ft = cost cosαt, and the second moment of F is σ 2 = + α 2. Hence, the measure F n from Theorem. represents the distribution of Z n = α + α 2 Z n + + α 2 Z n, where Z and Z n are independent normalized sums of n independent copies of X and X. Put n α = F n Φ. Since P{Z n = 0} P{Z n = 0} P{Z n = 0} > c n, we necessarily have nα > c n absolute constant c > 0. On the other hand, Proposition 6.4 implies: with some Corollary 7.. If for some η, η R, then lim inf n ] n η log n η nα > 0, 7. n α = O n 2 2η log n η 7.2 with η = 2η + 2η. In turn, the latter relation implies 7. with η = η 2 + η. This is a more precise formulation of Corollary.2. Note that 7. is impossible for η = and η < 0 by Dirichlet s theorem, so that necessarily η = ma { 2η + 2, 0 } = 2η Similarly, 7.2 is impossible for η = and η < 0. Relation 7. with η =, η = 0 defines the class of so-called badly approimable numbers α which can be characterized in terms of continued fractions. Namely, representing α = a 0 + a + a 2 + a where a 0 is an integer and a, a 2,... are positive integers, the property of being badly approimable is equivalent to i a i <. In particular, all quadratic irrationalities e.g. α = 2 belong to this class, cf. S2]. Since in this case η = /2, we arrive at: Corollary 7.2. For any badly approimable number α, we have n α = O n log n. It is not clear at all whether one can improve this rate for at least one α. On the other hand, at the epense of a logarithmic term, one may involve almost all values of α. To this aim, one may apply a theorem due to Khinchine which asserts the following cf. C], S2].,

17 7 Suppose that a function ψn > 0 is defined on the positive integers. If ψn is non-increasing and n= ψn =, then the inequality α p < ψn 7.3 n n has infinitely many integer solutions p, n for almost all α with respect to the Lebesgue measure on the real line. But when n= ψn <, 7.3 has only finitely many solutions for almost all α. This second assertion is an easy part of Khinchine s theorem, which may be quantified in terms of the function ] r ψ α = inf n ψn nα. Indeed, restricting ourselves without loss of generality to the values 0 < α <, first note that, for any integer n and δ 0, /2], mes{α 0, : nα < δ} = 2δ. Using this with δ = min{rψn, /2}, we get, for any r > 0, Therefore, mes{α 0, : ε ψ α < r} n= { mes α 0, : } ψn nα < r mes{α 0, : r ψ α < r} Cr r > 0 2rψn. with constant C = 2 n= ψn. In particular, r ψα > 0 for almost all α. For eample, choosing the sequence ψn = /n log +ε n +, Corollary 7. provides a rate which is applicable to almost all α. Corollary 7.3. Given ε > 0, for almost all α R, we have n α = O n log n3/2+ε. Perhaps, the power of the logarithmic term may be improved. At least, this is true on average when α varies inside a given interval, say 0 < α <. Proposition 7.4. With some absolute constant c > 0, for all n, 0 n α dα c n= logn n Proof. Our basic tool is the Berry-Esseen inequality of Lemma 2.2. For the distribution F, we have α 3 = EX 3 = 0 and β 4 = EX 4 = E X + αx 4 = + 6α 2 + α 4.

18 8 In order to control the integral in 2.3, note that σ 4 β 4 2σ 4. Using Lemma 2. with T = n gives that c n α n + I nα, where I n α = By simple calculus, for any t /2, ψ n t t 0 n /2 coss n ds 6t n, σ β4 2+α 2 2, cost cosαt n dt. 7.5 t so I n α dα = 0 n /2 cos t n t 2 ψ n t dt 6 n cos t n dt n t Thus, integrating the inequality in 7.5 over α, we are led to 7.4. /2 c logn +. n Remarks. Corollary 7.3 with quantity n α = F n Φ 3 remains to hold in a more general situation X = X + αx, where X, X are independent random variables with non-degenerate distributions and finite 4-th absolute moments. This etension requires an etra analysis of the behavior of characteristic functions, and we will discuss it somewhere else. Let us note that it is possible to improve the rate of convergence in particular, to remove the logarithmic term in models such as X = X 0 +α X + +α m X m with m 3 independent summands X k. See also K-S] on randomized versions of the central limit theorem. References B-B-S] Becher, V; Bugeaud, Y.; Slaman, T. A. The irrationality eponents of computable numbers. Proc. Amer. Math. Soc , no. 4, B] C] Ch] E] K-S] Beck, J. Probabilistic Diophantine approimation. Randomness in lattice point counting. Springer Monographs in Mathematics. Springer, Cham, 204, vi+487 pp. Cassels, J. W. S. An introduction to Diophantine approimation. Cambridge Tracts in Mathematics and Mathematical Physics, no. 45. Cambridge Univ. Press, NY, 957, +66 pp. Chistyakov, G. P. A new asymptotic epansion and asymptotically best constants in Lyapunov s theorem. I, II, III. Theory Probab. Appl , no. 2, ; , no. 3, ; , no. 3, Esseen, C-G. Fourier analysis of distribution functions. A mathematical study of the Laplace- Gaussian law. Acta Math , 25. Klartag, B.; Sodin, S. Variations on the Berry-Esseen theorem. Russian summary Teor. Veroyatn. Primen , no. 3, ; reprinted in: Theory Probab. Appl , no. 3, K] Kolodyazhnyi, S. F. Generalization of one theorem by Esseen. Russian Vestnik LGU, 3 968,

19 9 K-N] Kuipers, L.; Niederreiter, H. Uniform distribution of sequences. Pure and Applied Mathematics. Wiley-Interscience John Wiley & Sons], New York-London-Sydney, 974. iv+390 pp. P] Petrov, V. V. Sums of independent random variables. Translated from the Russian by A. A. Brown. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. Springer-Verlag, New York-Heidelberg, pp. Russian ed.: Moscow, Nauka, 972, 44 pp. P2] S] S2] Petrov, V. V. Limit theorems for sums of independent random variables. Russian Moscow, Nauka, 987, 320 pp. Schmidt, W. M. Diophantine approimation. Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 980. Schmidt, W. M. Diophantine approimations and Diophantine equations. Lecture Notes in Math. 467, Springer-Verlag, 99.