We prove a Carbery-Wright style anti-concentration inequality for the unitary Haar measure, by showing that the probability of a polynomial in the entries of a random unitary falling into an $\varepsilon$ range is at most a polynomial in $\varepsilon$. Using it, we show that the scrambling speed of a random ... more >>>

One of the most famous TFNP subclasses is PPP, which is the set of all search problems whose totality is guaranteed by the pigeonhole principle. The author's recent preprint [ECCC TR24-002 2024] has introduced a TFNP problem related to the pigeonhole principle over a quotient set, called Quotient Pigeon, and ... more >>>

We consider read-$k$ determinantal representations of polynomials and prove some non-expressibility results. A square matrix $M$ whose entries are variables or field elements will be called \emph{read-$k$}, if every variable occurs at most $k$ times in $M$. It will be called a \emph{determinantal representation} of a polynomial $f$ if $f=\det(M)$. ... more >>>