  The aim of this paper is to estimate the density f of a random variable X
when one has access to independent observations of the sum of K $\ge$ 2
independent copies of X. We provide a constructive estimator based on a
suitable definition of the logarithm of the empirical characteristic
function.We propose a new strategy for the data driven choice of the cut-off
parameter. The adaptive estimator is proven to be minimax-optimal up to some
logarithmic loss. A numerical study illustrates the performances of the method.
Moreover, we discuss the fact that the definition of the estimator applies in a
wider context than the one considered here.
