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We study the Hausdorff dimension of self-similar sets and measures on theline. We show that if the dimension is smaller than the minimum of 1 and thesimilarity dimension, then at small scales there are super-exponentially closecylinders. This is a step towards the folklore conjecture that such a drop indimension is explained only by exact overlaps, and confirms the conjecture incases where the contraction parameters are algebraic. It also gives anaffirmative answer to a conjecture of Furstenberg, showing that the projectionsof the "1-dimensional Sierpinski gasket" in irrational directions are all ofdimension 1.  As another consequence, if a family of self-similar sets or measures isparametrized in a real-analytic manner, then, under an extremely mildnon-degeneracy condition, the set of "exceptional" parameters has Hausdorffdimension 0. Thus, for example, there is at most a zero-dimensional set ofparameters 1/2<r<1 such that the corresponding Bernoulli convolution hasdimension <1, and similarly for Sinai's problem on iterated function systemsthat contract on average.  A central ingredient of the proof is an inverse theorem for the growth ofShannon entropy of convolutions of probability measures. For the dyadicpartition D_n of the line into intervals of length 1/2^n, we show that ifH(nu*mu,D_n)/n < H(mu,D_n)/n + delta for small delta and large n, then, whenrestricted to random element of a partition D_i, 0<i<n, either mu is close touniform or nu is close to atomic. This should be compared to results inadditive combinatorics that give the global structure of measures satisfyingH(nu*mu,D_n)/n < H(mu,D_n)/n + O(1/n).

On self-similar sets with overlaps and inverse theorems for entropy