On Fourier analytic properties of graphs
Jonathan M. Fraser, Tuomas Orponen, Tuomas Sahlsten
TL;DR
This paper investigates the Fourier dimensions of graphs of functions on [0,1], revealing that fractional Brownian motion graphs are not Salem sets and typical continuous functions have Fourier dimension zero.
Contribution
It provides new insights into the Fourier dimensions of function graphs, answering a longstanding question about fractional Brownian motion and typical functions.
Findings
Fractional Brownian motion graphs are almost surely not Salem sets.
Typical continuous functions have Fourier dimension zero.
The results partially answer Kahane's 1993 question.
Abstract
We study the Fourier dimensions of graphs of real-valued functions defined on the unit interval [0,1]. Our results imply that the graph of the fractional Brownian motion is almost surely not a Salem set, answering in part a question of Kahane from 1993, and that the graph of a Baire typical function in C[0,1] has Fourier dimension zero.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Mathematical Dynamics and Fractals