Hölder continuity and dimensions of fractal Fourier series
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| Formato: | Preprint |
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2022
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| _version_ | 1866912425815572480 |
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| author | Garitsis, Efstathios Konstantinos Chrontsios Hildebrand, AJ |
| author_facet | Garitsis, Efstathios Konstantinos Chrontsios Hildebrand, AJ |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_09806 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hölder continuity and dimensions of fractal Fourier series Garitsis, Efstathios Konstantinos Chrontsios Hildebrand, AJ Classical Analysis and ODEs Metric Geometry Number Theory 28A80, 11M26, 26A16, 42A16 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. |
| title | Hölder continuity and dimensions of fractal Fourier series |
| topic | Classical Analysis and ODEs Metric Geometry Number Theory 28A80, 11M26, 26A16, 42A16 |
| url | https://arxiv.org/abs/2208.09806 |