  We prove two basic conjectures on the distribution of the smallest singular
value of random n times n matrices with independent entries. Under minimal
moment assumptions, we show that the smallest singular value is of order
n^{-1/2}, which is optimal for Gaussian matrices. Moreover, we give a optimal
estimate on the tail probability. This comes as a consequence of a new and
essentially sharp estimate in the Littlewood-Offord problem: for i.i.d. random
variables X_k and real numbers a_k, determine the probability P that the sum of
a_k X_k lies near some number v. For arbitrary coefficients a_k of the same
order of magnitude, we show that they essentially lie in an arithmetic
progression of length 1/p.
