We derive an accurate lower tail estimate on the lowest singular value $\sigma_1(X-z)$ of a real Gaussian (Ginibre) random matrix $X$ shifted by complex parameter $z$. Such shift effectively changes upper behaviour condition number $\kappa(X-z)$ from slower $\mathbf{P}(\kappa(X-z)\ge t)\lesssim 1/t$ decay typical for Ginibre matrices to faster $1/t^2$ seen as long $z$ is away axis. This sharpen...

The distribution of $$\alpha p$$ modulo one, where p runs over the rational primes and $$ is a fixed irrational real, has received lot attention. It natural to ask for which exponents $$\nu >0$$ one can establish infinitude satisfying $$||\alpha p||\le p^{-\nu }$$ . latest record in this regard Kaisa Matomäki’s landmark result =1/3-\varepsilon presents limit currently known technology. Recently...