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Let $N(\alpha, n)$ denote the number of integer points inside the origin-centered ball of radius $\alpha \sqrt n$ in $n$ dimensions, where $\alpha \in (0,\infty)$ is some fixed constant. Precisely:

$$N(\alpha, n) := |\{z\in \mathbb{Z}^n:\|z\|_2 \le \alpha \sqrt n\}|.$$

I am curious about what is known regarding explicit formulae for: $$\lim_{n \to \infty} \frac 1 n\log N(\alpha, n). $$

There is a classical paper of J.E. Mazo and A.M. Odlyzko (1990) called "Lattice points in high dimensional spheres". This paper gives an explicit asymptotic in equation 3.4, but it is given in terms of fixed points of some transcendental equations.

It seems sort of natural that this is the case. Nonetheless, I am wondering: are any alternate derivations or identities known for this asymptotic?

Some concrete examples of what I am looking for:

  • Are there any specific (non-trivial) values of $\alpha$ for which this formula can be evaluated/understood?
  • Are there any probablistic interpretations (since the discrete Gaussian seems to arise in the mentioned formula)?
  • Is there some natural “positive temperature” relaxation of the criterion $\|z\|_2 \le \alpha \sqrt n$ that lends itself to some heuristic computation via statistical physics methods (cavity or replica)?
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  • $\begingroup$ Link to publication version of paper: link.springer.com/article/10.1007/bf01571276 Link to open access PDF (unfortunately, poorly formatted): www-users.cse.umn.edu/~odlyzko/doc/arch/high.dim.spheres.pdf $\endgroup$
    – DJA
    Commented Sep 4 at 1:14
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    $\begingroup$ On pg. 2 of the linked paper, although they say they are defining a sphere, what they seem to define is a closed ball, no? $\endgroup$
    – Sam Hopkins
    Commented Sep 4 at 1:25
  • $\begingroup$ This is a relevant distinction for your problem because for most $\alpha$ (when the radius squared is not an integer) there obviously cannot be any lattice points on a sphere of that radius. $\endgroup$
    – Sam Hopkins
    Commented Sep 4 at 1:28
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    $\begingroup$ @LSpice well yes but there is also the factor of $\sqrt{n}$ which can cancel denominators in $\alpha$ sometimes. Anyways the question is better suited for balls than spheres and it has been modified appropriately. $\endgroup$
    – Sam Hopkins
    Commented Sep 4 at 12:44
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    $\begingroup$ It seems that there is at least one obvious probabilistic interpretation: Let $u_i$ be a sequence of iid uniforms on $\{-M,-M+1,\dots,0,\dots, M-1, M\}$ (where $M$ is some large integer that I will take to infinity later). Let $v_i = u_i^2$. Then, if one unpacks the formula in the linked paper, it is essentially Cramer's large deviation theorem applied to the $v_i$. Apologies for not checking this more carefully before posting the question. I am still interested in other interpretations, so I will leave the question open for a bit before posting this as an answer. $\endgroup$
    – DJA
    Commented Sep 4 at 13:04

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