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We study concentration properties of random vectors of the form $AX$, where$X = (X_1, ..., X_n)$ has independent coordinates and $A$ is a given matrix. Weshow that the distribution of $AX$ is well spread in space whenever thedistributions of $X_i$ are well spread on the line. Specifically, assume thatthe 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 $T\|A\|_{HS}$ is at most $(Cp)^{0.9 r(A)}$, where $r(A)$ denotes the stable rankof $A$ and $C$ is an absolute constant.

Small ball probabilities for linear images of high dimensional distributions