Additional results of Zalcman's Pompeiu problem (Q1195611)

The author considers the following covering problem posed by L. Zalcman. Let be \(S\) be the open unit square and for each \(z\in S\) let be \(S(z)\) the largest open square in \(S\) with centroid \(z\). Fix \(0<\alpha\leq 1\). Let \(S_ \alpha(z)\) the square homothetic to \(S(z)\) with ratio \(\alpha\). The question is, assume that \(f\in C(\overline {S})\) and \(\int_{S_ \alpha(z)} f(x)dx=0\) for every \(z\in S\), does it follow that \(f\equiv 0\)? Zalcman answered affirmatively for \(\alpha=1\), \(\alpha=1/2\). The author and T. Schonbek later extended this result to \(\alpha= n/(n+2)\), \(n\in\mathbb{N}^*\). In the current paper the author answers the question affirmatively for \(3/4\leq\alpha<1\).