New sign uncertainty principles

New sign uncertainty principles, Discrete Analysis 2023:9, 46 pp. The famous Heisenberg uncertainty principle is an inequality that states that the product of the variances of a function $f$ and its Fourier transform $\hat f$ is bounded below by an absolute constant, which has the well-known consequence that one cannot simultaneously know the position and momentum of a particle to an arbitrary degree of accuracy. This paper concerns a different kind of uncertainty principle, which is a little more complicated to state. Given a function $f:\mathbb R^d\to\mathbb R$, let us define the _positivity radius_ $r(f)$ of $f$ to be the smallest $r$ such that $|f(x)|\geq 0$ for every $x$ with $\|x\|\geq r$. Suppose now that $f\in L^1(\mathbb R^2)$ is an even function, which implies that $\hat f$ is real valued. If we assume that $f$ is not identically zero, and also that $\hat f(0)\leq 0$, then since $\hat f(0)$ is just the integral of $f$, it follows that $f$ must be negative on a set of positive measure, and hence that $r(f)>0$. Similarly, if we assume that $f(0)\leq 0$ (while still assuming that $f$ is not identically zero, which implies that $\hat f$ is not identically zero), then $r(\hat f)>0$. It was shown by Bourgain, Clozel and Kahane that $r(f)r(\hat f)$ is also bounded below by an absolute constant: that is, if both $f$ and $\hat f$ are non-negative for sufficiently large $x$, then the regions where they are negative cannot both be too concentrated about the origin. (Note that unlike with the Heisenberg uncertainty principle, the origin plays a special role here, since the assumption that $f$ is even is not translation invariant.) A variant of this result with a similar conclusion was discovered by Cohn and Gonçalves (the latter one of the authors of this paper) in which the inequalities for $\hat f$ are reversed: that is, $\hat f(0)\geq 0$ and $\hat f(x)\leq 0$ for all sufficiently large $x$. This variant is interesting because it relates to a result of Cohn and Elkies from 2003 that yields upper bounds for sphere packing. One formulation of their result is that if $f:\mathbb R^d\to\mathbb R$ is an "admissible" function (which means that $f$ and $\hat f$ both satisfy a suitable decay condition), $f(0)=\hat f(0)>0$, $f(x)\leq 0$ is positive for all $x$ with $\|x\|\geq r$, and $\hat f(x)\geq 0$ everywhere, then the density of sphere packings in $\mathbb R^d$ is bounded above by $(r/2)^d$. In a further spectacular development, Viazovska found a suitable functi