## Combinatorics Seminar - Spring '21

When: Sunday, May 16, 10am

Where: Schreiber 309

Speaker: Asaf Cohen
Antonir, Tel Aviv University

Title:
Exact Limit Theorems for Restricted Integer Partitions

## Abstract:

For a set of positive integers $A$, let $p_A(n)$ denote the number of ways
to
write $n$ as a sum of
integers from $A$, and let $p(n)$ denote the usual partition function. In
the
early 40s, Erdos extended
the classical Hardy-Ramanujan formula for $p(n)$ by showing that $A$ has
density $\alpha$ if and only if
$\log p_A(n) \sim \log p(\alpha n)$. Nathanson asked if Erdos's theorem
holds also
with respect to $A$'s lower
density, namely, whether $A$ has lower-density $\alpha$ if and only if
$\log p_A(n)/
\log p(\alpha n)$ has lower limit
1. We answer this question negatively by constructing, for every $\alpha >
0$, a
set of integers $A$ of
lower density $\alpha$, satisfying

$$
\liminf_{n \mapsto \infty}\frac{\log p_A(n)}{\log p(\alpha n)} \geq
\left(\frac{\sqrt{6}}{\pi}-o_{\alpha}(1)\right)\log(1/\alpha).
$$

We further show that the above bound is best possible (up to the
$o_{\alpha}(1)$
term), thus determining
the exact extremal relation between the lower density of a set of integers
and the lower limit of
its partition function. We also prove an analogous theorem with respect to
the upper density of
a set of integers, answering another question of Nathanson.

Joint work with Asaf Shapira.