  We study the structure and learnability of sums of independent integer random
variables (SIIRVs). For $k \in \mathbb{Z}_{+}$, a $k$-SIIRV of order $n \in
\mathbb{Z}_{+}$ is the probability distribution of the sum of $n$ independent
random variables each supported on $\{0, 1, \dots, k-1\}$. We denote by ${\cal
S}_{n,k}$ the set of all $k$-SIIRVs of order $n$.
  In this paper, we tightly characterize the sample and computational
complexity of learning $k$-SIIRVs. More precisely, we design a computationally
efficient algorithm that uses $\widetilde{O}(k/\epsilon^2)$ samples, and learns
an arbitrary $k$-SIIRV within error $\epsilon,$ in total variation distance.
Moreover, we show that the {\em optimal} sample complexity of this learning
problem is $\Theta((k/\epsilon^2)\sqrt{\log(1/\epsilon)}).$ Our algorithm
proceeds by learning the Fourier transform of the target $k$-SIIRV in its
effective support. Its correctness relies on the {\em approximate sparsity} of
the Fourier transform of $k$-SIIRVs -- a structural property that we establish,
roughly stating that the Fourier transform of $k$-SIIRVs has small magnitude
outside a small set.
  Along the way we prove several new structural results about $k$-SIIRVs. As
one of our main structural contributions, we give an efficient algorithm to
construct a sparse {\em proper} $\epsilon$-cover for ${\cal S}_{n,k},$ in total
variation distance. We also obtain a novel geometric characterization of the
space of $k$-SIIRVs. Our characterization allows us to prove a tight lower
bound on the size of $\epsilon$-covers for ${\cal S}_{n,k}$, and is the key
ingredient in our tight sample complexity lower bound.
  Our approach of exploiting the sparsity of the Fourier transform in
distribution learning is general, and has recently found additional
applications.
