The main purpose of this note is to estimate the size of the set Tμλ of points, at which the Cantor function is not differentiable and we find that the Hausdorff dimension of Tμλ is [log2/log3]2. Moreover, the Pac...The main purpose of this note is to estimate the size of the set Tμλ of points, at which the Cantor function is not differentiable and we find that the Hausdorff dimension of Tμλ is [log2/log3]2. Moreover, the Packing dimension of Tμλ is log2/log3. The log2 = loge2 is that if ax = N (a >0, and a≠1), then the number x is called the logarithm of N with a base, recorded as x = logaN, read as the logarithm of N with a base, where a is called logarithm Base number, N is called true number.展开更多
The existence of large partial quotients destroys many limit theorems in the metric theory of continued fractions.To achieve some variant forms of limit theorems,a common approach mostly used in practice is to discard...The existence of large partial quotients destroys many limit theorems in the metric theory of continued fractions.To achieve some variant forms of limit theorems,a common approach mostly used in practice is to discard the largest partial quotient,while this approach works in obtaining limit theorems only when there cannot exist two terms of large partial quotients in a metric sense.Motivated by this,we are led to consider the metric theory of points with at least two large partial quotients.More precisely,denoting by[a1(x),a2(x),...]the continued fraction expansion of x∈[0,1)and lettingψ:N→R+be a positive function tending to in nity as n→∞,we present a complete characterization on the metric properties of the set,i.e.,E(ψ)={x∈[0,1):∃16 k̸=ℓ6 n,ak(x)>ψ(n),aℓ(x)>ψ(n)for in nitely many n∈N}in the sense of the Lebesgue measure(the Borel-Bernstein type result)and the Hausdor dimension(the Jarnik type result).The main result implies that any nite deletion from a1(x)+……+an(x)cannot result in a law of large numbers.展开更多
文摘The main purpose of this note is to estimate the size of the set Tμλ of points, at which the Cantor function is not differentiable and we find that the Hausdorff dimension of Tμλ is [log2/log3]2. Moreover, the Packing dimension of Tμλ is log2/log3. The log2 = loge2 is that if ax = N (a >0, and a≠1), then the number x is called the logarithm of N with a base, recorded as x = logaN, read as the logarithm of N with a base, where a is called logarithm Base number, N is called true number.
基金supported by National Natural Science Foundation of China(Grant Nos.12171172 and 11831007)。
文摘The existence of large partial quotients destroys many limit theorems in the metric theory of continued fractions.To achieve some variant forms of limit theorems,a common approach mostly used in practice is to discard the largest partial quotient,while this approach works in obtaining limit theorems only when there cannot exist two terms of large partial quotients in a metric sense.Motivated by this,we are led to consider the metric theory of points with at least two large partial quotients.More precisely,denoting by[a1(x),a2(x),...]the continued fraction expansion of x∈[0,1)and lettingψ:N→R+be a positive function tending to in nity as n→∞,we present a complete characterization on the metric properties of the set,i.e.,E(ψ)={x∈[0,1):∃16 k̸=ℓ6 n,ak(x)>ψ(n),aℓ(x)>ψ(n)for in nitely many n∈N}in the sense of the Lebesgue measure(the Borel-Bernstein type result)and the Hausdor dimension(the Jarnik type result).The main result implies that any nite deletion from a1(x)+……+an(x)cannot result in a law of large numbers.
