University of Birmingham > Talks@bham > Topology and Dynamics Seminar > Uniqueness of trigonometric series outside fractals

Uniqueness of trigonometric series outside fractals

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A subset F of [0,1] is called a set of uniqueness if trigonometric series are unique outside of F. Otherwise F is called a set of multiplicity. The uniqueness problem in harmonic analysis dating back to the fundamental works of Riemann, Cantor et al. concerns about classifications of sets of uniqueness and multiplicity. Typically one expects the sets of uniqueness to have ’non-chaotic’ / orderly features and where as the sets of multiplicity should be `chaotic’. This is highlighted by the theorem of Piatetski-Shapiro-Salem-Zygmund (1954) stating that middle lambda-Cantor sets is a set of uniqueness if and only if 1/lambda is a Pisot number (roughly speaking numbers whose powers approximate integers at an exponential rate).

In our work we attempt to characterise the multiplicity or uniqueness of a fractal F using the statistical/dynamical properties of the fractal F. In particular, we prove any non-lattice self-similar set is a set of multiplicity. We also establish an analogous result in higher dimensions for self-affine sets, where the non-lattice condition is replaced by the irreducibility of the subgroup defined by the linear parts of the contractions. The statistical theory we use is the renewal theory for random walks on Lie groups.

Joint work with Jialun Li (Bordeaux & Zürich).

This talk is part of the Topology and Dynamics Seminar series.

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