Small ball probabilities for linear images of high dimensional distributions

We study concentration properties of random vectors of the form $AX$, where $X=(X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n)$ has independent coordinates and $A$ is a given matrix. We show that the distribution of $AX$ is well spread in space whenever the distributions of $X_i$ are well spread on the line. Specifically, assume that the probability that $X_i$ falls in any given interval of length $T$ is at most $p$. Then the probability that $AX$ falls in any given ball of radius $T |A|_{HS}$isatmost$(Cp)^{0.9 r(A)}$,where$r(A)$ denotes the stable rank of $A$ and $C$ is an absolute constant.