论文标题
Hölder连续性和分形傅立叶系列的尺寸
Hölder continuity and dimensions of fractal Fourier series
论文作者
论文摘要
由数字理论，分析和分形几何形状中的应用激励，我们考虑了与形式$ f（t）= \ sum_ {n = 1}^\ infty f（n）e^{2πint}/n $相关的傅立叶系列相关的图形和尺寸。我们的主要结果指出，如果对于某些常数$ c $和$α$，则$ 0 <α<1 $，我们有$ | \ sum_ {1 \ le n \ le n \ le x} f（n）e^{2πint} | \ le c x^α$ c x^α$在$ x \ ge 1 $和$ x $ x \ ge in \ in \ in \ in \ in \ in \ in \ in \ in \ in \ in \ in c $ co.连续与指数$ 1-α$连续，以及$ | f（t）| $的图表$ [0,1] $具有盒子计数尺寸$ \ leq 1+α$。作为应用程序，我们恢复了Weierstrass功能的最佳能力统一Hölder指数$ \ sum_ {k = 1}^\ infty a^k \ cos（2πb^k t）$和riemann函数$ \ sum_ {n = 1}^}^\ infty \ sin（iffty \ sin（infty \ sin（sin（sunfty））\ sin（sin（sin^n^2 t）/n^2 $。在普遍的Riemann假设的假设下，我们获得了Hölder指数和尺寸与一系列$ \ sum_ {n = 1}^\inftyμ（n）e^{2πin^kt}/n^k $，其中$ \ undty n^k $ $ $ $ uip是mouim uip formict。
Motivated by applications in number theory, analysis, and fractal geometry, we consider regularity properties and dimensions of graphs associated with Fourier series of the form $F(t)=\sum_{n=1}^\infty f(n)e^{2πi nt}/n$, for a large class of coefficient functions $f$. Our main result states that if, for some constants $C$ and $α$ with $0<α<1$, we have $|\sum_{1\le n\le x}f(n)e^{2πi nt}|\le C x^α$ uniformly in $x\ge 1$ and $t\in \mathbb{R}$, then the series $F(t)$ is Hölder continuous with exponent $1-α$, and the graph of $|F(t)|$ on the interval $[0,1]$ has box-counting dimension $\leq 1+α$. As applications we recover the best-possible uniform Hölder exponents for the Weierstrass functions $\sum_{k=1}^\infty a^k\cos(2πb^k t)$ and the Riemann function $\sum_{n=1}^\infty \sin(πn^2 t)/n^2$. Moreoever, under the assumption of the Generalized Riemann Hypothesis, we obtain nontrivial bounds for Hölder exponents and dimensions associated with series of the form $\sum_{n=1}^\infty μ(n)e^{2πi n^kt}/n^k$, where $μ$ is the Möbius function.
