# Explicit Salem Sets in R^n

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.