Spectral synthesis with the complexity parameter in $\mathbb{Z}_N^d$

We study asymptotic spectral synthesis on $\mathbb Z_N^d$ using the Fourier ratio, the quotient of the counting-measure $\ell^1$ and $\ell^2$ norms of the unitary Fourier transform. For supports of cardinality $N^{α+o(1)}$, the Fourier-ratio concentration exponent satisfies $0\leqκ\leq\fracα{2}$. We