The Shubin--Vakilian--Wolff Uncertainty Principle at Half Density

Shubin, Vakilian, and Wolff proved a Fourier uncertainty principle for sets of sufficiently small local density at the reciprocal scale and asked whether every density below one is admissible. We answer this question negatively in dimension one by constructing sequences of pairs of $1\mathrm{/}2$-density thin sets and unit vectors whose total position and Fourier mass outside these sets tends to zero. This obstruction persists for every scaled reciprocal profile ${\rho }_{\kappa }(x)=\min \{1,\kappa \mathrm{/}\mathrm{\mid }x\mathrm{\mid }\}$, $\kappa >0$. On the other hand, for $0<\kappa \le 1$, the uncertainty estimate holds whenever the density $$\epsilon <\frac{1}{2(1+32\kappa )}\mathrm{.}$$ Thus the critical density tends to $1\mathrm{/}2$ as $\kappa \downarrow 0$. The obstruction uses odd Gaussian packets to transfer norm bounds from a free-group model. The positive estimate uses a Fourier-complementary anti-Wick operator and quadratic straightening of the reciprocal geometry.