How few three-term arithmetic progressions can a subset $S \subseteq \Z_N := \Z/N\Z$ have if $|S| \geq \upsilon N$ (that is, $S$ has density at least $\upsilon$)? Varnavides %\cite{varnavides} showed that this number of arithmetic progressions is at least $c(\upsilon)N^2$ for sufficiently large integers $N$. It is well known that determining good lower bounds for $c(\upsilon)> 0$ is at the same level of depth as Erd\" os's famous conjecture about whether a subset $T$ of the naturals where $\sum_{n \in T} 1/n$ diverges, has a $k$-term arithmetic progression for $k=3$ (that is, a three-term arithmetic progression). We answer a question posed by B. Green %\cite{AIM} about how this minimial number of progressions oscillates for a fixed density $\upsilon$ as $N$ runs through the primes, and as $N$ runs through the odd positive integers.

MSC Classifications:

05D99 show english descriptions None of the above, but in this section 05D99 - None of the above, but in this section

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