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The Minimal Number of ThreeTerm Arithmetic Progressions Modulo a Prime Converges to a Limit


Published:20080301
Printed: Mar 2008
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Abstract
How few threeterm 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 threeterm 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.