Abstract: The study of the most natural laws suggest that if integral valued random variables X_1, ... , X_n, ... have a large tail (e.g., Pr{|X|>L}>>L^(-a) for every L>2 and some a<2), then the concentration of the sum S_n = X_1 + ... + X_n should be small (e.g., Pr{S_n=k}<n^(-1/a), uniformly in k), at least when those random variables satisfy the usual requirement to be independant and identically distributed. Using Fourier transform and tools from the inverse additive number theory, Freiman, Yudin and the speaker showed that this is indeed the case (with a constant independent of the law of the X's) if a is not too close to 0 and the X's satisfy a certain arithmetic condition. Sutanto and the speaker have recently shown that this arithmetic condition cannot be dispensed with, and have further shown that this condition is not sufficient when a is too close to 0.