A uniform set with fewer than expected arithmetic progressions of length 4

Abstract

An example is presented of a subset $A$ of ${\mathbb{Z}}_N$ of density $\alpha$ such that the largest non-trivial Fourier coefficient of the characteristic function of $A$ is very small, but the probability that a random arithmetic progression (mod $N$) of length 4 lies in $A$ is significantly smaller than ${\alpha }^4$.

This preprint was written well over a decade ago and has been known about by experts but not made publicly available. I apologize for this and am now remedying the situation. Revised version updates the notation, corrects minor errors (pointed out by referee), and adds one or two references

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