Can we get an upper bound on the number of integral points satisfying the following inequality:-
$x_1^p + x_2^p + ... + x_k^p < N$ where $N$ is some real number, $p\leq 1$ and $0\leq x_i \leq D$.
A simple approach that I can think of is to bound the number of points with the area underneath by the curve. eg. for $p = 1$, volume bounded by the solid is $\frac{N^k}{k!}$. This is basically the volume of a simplex with equal intercepts.