  An $(n, k)$-Poisson Multinomial Distribution (PMD) is a random variable of
the form $X = \sum_{i=1}^n X_i$, where the $X_i$'s are independent random
vectors supported on the set of standard basis vectors in $\mathbb{R}^k.$ In
this paper, we obtain a refined structural understanding of PMDs by analyzing
their Fourier transform. As our core structural result, we prove that the
Fourier transform of PMDs is {\em approximately sparse}, i.e., roughly
speaking, its $L_1$-norm is small outside a small set. By building on this
result, we obtain the following applications:
  {\bf Learning Theory.} We design the first computationally efficient learning
algorithm for PMDs with respect to the total variation distance. Our algorithm
learns an arbitrary $(n, k)$-PMD within variation distance $\epsilon$ using a
near-optimal sample size of $\widetilde{O}_k(1/\epsilon^2),$ and runs in time
$\widetilde{O}_k(1/\epsilon^2) \cdot \log n.$ Previously, no algorithm with a
$\mathrm{poly}(1/\epsilon)$ runtime was known, even for $k=3.$
  {\bf Game Theory.} We give the first efficient polynomial-time approximation
scheme (EPTAS) for computing Nash equilibria in anonymous games. For normalized
anonymous games with $n$ players and $k$ strategies, our algorithm computes a
well-supported $\epsilon$-Nash equilibrium in time $n^{O(k^3)} \cdot
(k/\epsilon)^{O(k^3\log(k/\epsilon)/\log\log(k/\epsilon))^{k-1}}.$ The best
previous algorithm for this problem had running time $n^{(f(k)/\epsilon)^k},$
where $f(k) = \Omega(k^{k^2})$, for any $k>2.$
  {\bf Statistics.} We prove a multivariate central limit theorem (CLT) that
relates an arbitrary PMD to a discretized multivariate Gaussian with the same
mean and covariance, in total variation distance. Our new CLT strengthens the
CLT of Valiant and Valiant by completely removing the dependence on $n$ in the
error bound.
