Small Fourier Wiener norm for arbitrarily large 3-AP-free sets

Determine whether, for every epsilon>0, there exist arbitrarily large 3-term-arithmetic-progression-free sets AsubsetZ whose Fourier L norm satisfies A|A|^\epsilon.

Background

Theorem 3-AP proves that a sufficiently small Fourier L norm prevents a large subset of A from being 3-term-arithmetic-progression-free, establishing a threshold of the form exp((log |A|)c). The authors ask whether this threshold can be substantially improved to polynomially small Fourier norm, namely |A|\epsilon for every fixed positive \epsilon.

They discuss the computational example A_N={sum_{i=0}{N-1} d_i4i:d_iin{0,1}}, which is 3-term-arithmetic-progression-free and appears to satisfy a bound with exponent below 0.3334 for N=9. They also compare it with the two-point example K={n,n+1}, whose exponent is approximately 0.3485.

References

An interesting open question, probing the sharpness of the threshold in Theorem \ref{thm:3-AP}, is the following: given $\epsilon>0$, can one construct a large 3-AP-free set $A\subset \mathbb{Z}$ such that $|A|\leq |A|\epsilon$? A non-exhaustive computer search gives that the set $A_N=\Big{\sum_{i=0}{N-1} d_i4i:d_i\in {0,1}\Big}$ is 3-AP-free and that $|A_N|\leq |A_N|\epsilon$ with $\epsilon< 0.3334$ for $N=9$, say. This beats the trivial example of the two point set $K={n,n+1}$, which satisfies $|K|=|K|{\delta}$ with $\delta = \frac{\log(4/\pi )}{\log 2}> 0.3485$. It remains open whether arbitrarily small positive exponents are possible.

— Remarks on the Geometry of Sets in $\mathbb{Z}^d$ with Small Fourier $L^1$ Norm  (2609.26619 - Burgin et al., 22 Sep 2026) in Section 3, immediately following the corollary on Weyl sums