Massive Helson Sets

Student: Yanina Anastasiya

Faculty: HSE Tikhonov Moscow Institute of Electronics and Mathematics (MIEM HSE)

Educational Programme: Applied Mathematics (Bachelor)

Year of Graduation: 2019

We consider the Helson sets, i.e., compact subsets $E$ of the torus $\mathbb T^n$, such that every continuous function on $E$ extends to a function that belongs to the Wiener algebra of absolutely convergent Fourier series. Wik's theorem states the existence of Helson set of Hausdorff dimension 1 on the circle. We obtain a similar result in the multidimensional case, namely, we show that there exists an uncountable Helson set $E \subset \mathbb{T}^n$ $(n \geq 2)$ satisfying $\dim_M E = n.$

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