The Zero Set of Fractional Brownian Motion Is a Salem Set

The existence of sets supporting a Borel measure such that its Fourier transform tends to zero at infinity can be traced back to the problem of uniqueness of trigonometric series, studied extensively by Cantor. Given α∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0, 1)$$\end{document}, Beurling asked if there exists a subset of the real line of Hausdorff dimension α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} supporting a Borel measure whose Fourier transform converges to zero at infinity with rate α/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha /2$$\end{document}. Salem answered the question in the affirmative and such sets are now called Salem sets or rounded sets. Kahane showed that images of compact sets by fractional Brownian motion are Salem sets and this was recently extended to Gaussian random fields with stationary increments and to multi-parameter Brownian sheets. He asked if the level sets of fractional Brownian motion are also Salem sets and the problem has remained open since. This paper answers Kahane’s question in the affirmative. The argument is based on the study of oscillatory integrals with non-smooth amplitudes and new properties of the generalised Euler spiral which have independent interest.