Maximum entropy Edgeworth estimates of the number of integer points in polytopes

Understand

Abstract: The number of points $x=(x_1 ,x_2 ,...x_n)$ that lie in an integer cube $C$ in $R^n$ and satisfy the constraints $\sum_j h_{ij}(x_j )=s_i ,1\le i\le d$ is approximated by an Edgeworth-corrected Gaussian formula based on the maximum entropy density $p$ on $x \in C$, that satisfies $E\sum_j h_{ij}(x_j )=s_i ,1\le i\le d$.

• Under $p$, the variables $X_1 ,X_2 ,...X_n $ are independent with densities of exponential form.
• Letting $S_i$ denote the random variable $\sum_j h_{ij}(X_j )$, conditional on $S=s, X$ is uniformly distributed over the integers in $C$ that satisfy $S=s$.
• The number of points in $C$ satisfying $S=s$ is $p \{S=s\}\exp (I(p))$ where $I(p)$ is the entropy of the density $p$.