  We introduce a general framework for studying anticoncentration and local
limit theorems for random variables, including graph statistics. Our methods
involve an interplay between Fourier analysis, decoupling, hypercontractivity
of Boolean functions, and transference between ``fixed-size'' and
``independent'' models. We also adapt a notion of ``graph factors'' due to
Janson.
  As a consequence, we derive a local central limit theorem for connected
subgraph counts in the Erd\H{o}s-Renyi random graph $G(n,p)$, building on work
of Gilmer and Kopparty and of Berkowitz. These results improve an
anticoncentration result of Fox, Kwan, and Sauermann and partially answers a
question of Fox, Kwan, and Sauermann. We also derive a local limit central
limit theorem for induced subgraph counts, as long as $p$ is bounded away from
a set of ``problematic'' densities, partially answering a question of Fox,
Kwan, and Sauermann. We then prove these restrictions are necessary by
exhibiting a disconnected graph for which anticoncentration for subgraph counts
at the optimal scale fails for all constant $p$, and finding a graph $H$ for
which anticoncentration for induced subgraph counts fails in $G(n,1/2)$. These
counterexamples resolve anticoncentration conjectures of Fox, Kwan, and
Sauermann in the negative.
  Finally, we also examine the behavior of counts of $k$-term arithmetic
progressions in subsets of $\mathbb{Z}/n\mathbb{Z}$ and deduce a local limit
theorem wherein the behavior is Gaussian at a global scale but has nontrivial
local oscillations (according to a Ramanujan theta function). These results
improve on results of and answer questions of the authors and Berkowitz, and
answer a question of Fox, Kwan, and Sauermann.
