Proof of a conjecture on the supports of Wigner distributions (Q1282025)

The author proves that if \(f\neq 0\) belongs to \(L^2(\mathbb R^n)\) then the set of nonvanishing of its Wigner distribution \(W(f)\) is infinite. The result was conjectured by Mustard and Sitaram by analogy with a theorem of Benedicks saying that if \(f\in L^1\) and \(\widehat f\) are both nonvanishing only on sets of finite measure then \(f\equiv 0\). In fact, the result is proved by applying Benedicks' theorem after a clever use of Moyal's formula. A similar result is proved when \(W(f)\) is restricted to a half space. The author notes that the conjecture was proved independently by P. Jaming and E. Wilczok.