Explicit Salem Sets in R^n

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• Rob Fraser (Edinburgh)
• Tuesday 21 January 2020, 14:00-15:00
• WATN-R17 18, Watson Building.

If you have a question about this talk, please contact Hong Duong.

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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• Analysis seminar
• School of Mathematics events
• WATN-R17 18, Watson Building

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