University of Birmingham > Talks@bham > Analysis Seminar > Explicit Salem Sets in R^n

Explicit Salem Sets in R^n

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In 1981, R. Kaufman showed that the $\tau$-approximable numbers in $\mathbb{R}$ support a measure $\mu$ satisfying

$ |\hat{\mu}|\leq C_\epsilon (1+|\xi|) -1/(1+\tau)+\epsilon $

for any $\epsilon>0$. The exponent $-1/(1+\tau)$ is optimal for a set with Hausdorff dimension $2/(1+\tau)$. Sets supporting a measure with nearly optimal pointwise Fourier decay for their Hausdorff dimension are called Salem sets. We show that a higher-dimensional analogue of the $\tau$-approximable numbers is a Salem set. This provides the first explicit example of a Salem set of dimension other than 0, n-1, or n in $\mathbb{R}n$. (Joint work with Kyle Hambrook)

This talk is part of the Analysis Seminar series.

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