# Probability of sparse spectrum

Consider a vector $$v$$ such that $$v \sim \mathrm{Unif}(\mathbb{S}^{d-1})$$, the uniform distribution on the unit sphere in $$d$$ dimensions.

Question: is there an upper bound on the probability that $$v$$ has a sparse spectrum? (e.g. $$\mathbb{P}\left( |\mathrm{supp}(\hat{v})| \leq \sqrt{d}\right)$$, where $$\hat{v}$$ is the DFT of $$v$$)? Any pointers to references are more than welcome.

• What is g hat for? – Carlos Campos May 24 '19 at 23:53
• @CarlosCampos: Sorry, that was a typo. $\hat{v}$ is the DFT of the vector $v$. – VHarisop May 24 '19 at 23:54