论文标题
自相似措施的等分分配结果
Equidistribution results for self-similar measures
论文作者
论文摘要
由于Koksma而导致的一个众所周知的定理指出，对于Lebesgue而言，几乎每$ x> 1 $ the sequence $（x^n）_ {n = 1}^{\ infty} $是均匀分布的模量。在本文中，我们给出了足够的条件，使该定理的类似物以进行自相似措施。我们的方法更普遍地适用于$（f_ {n}（x））_ {n = 1}^{\ infty} $的序列，其中$（f_n）_ {n = 1}^{\ infty} $是一个充分平滑的真实值函数的序列，使得满足非线性的假设。作为我们的主要结果的必然性，我们表明，如果$ c $等于中间第三个cantor set和$ t \ geq 1 $，那么就$ c+t $上的cantor-lebesgue量度而言，几乎每个$ x $都均匀地分布在$ c+t $上。
A well known theorem due to Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In this paper we give sufficient conditions for an analogue of this theorem to hold for self-similar measures. Our approach applies more generally to sequences of the form $(f_{n}(x))_{n=1}^{\infty}$ where $(f_n)_{n=1}^{\infty}$ is a sequence of sufficiently smooth real valued functions satisfying a nonlinearity assumption. As a corollary of our main result, we show that if $C$ is equal to the middle third Cantor set and $t\geq 1$, then with respect to the Cantor-Lebesgue measure on $C+t$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed for almost every $x$.
