Uniqueness for spherically convergent multiple trigonometric series (Q2702478)

This is a chapter in the book Analytic-Computational Methods and is devoted to multidimensional extensions of Cantor's theorem. The latter asserts that if a trigonometric series converges to \(0\) at every point of \([-\pi,\pi)\), then its sum is \(0\) identically. The author formulates four potential generalizations of this theorem. NEWLINENEWLINENEWLINETheorem 7.2. Suppose that \(S(x)=\sum_{n\in\mathbf Z^d}c_n e^{inx}\) converges unrestrictedly rectangularly to \(0\) at every point of \(\mathbf T^d\). Then \(S\equiv 0\). NEWLINENEWLINENEWLINEConjecture 7.1. Suppose that \(S(x)=\sum_{n\in\mathbf Z^d}c_n e^{inx}\) converges squarely to \(0\) at every point of \(\mathbf T^d.\) Then \(S\equiv 0\). NEWLINENEWLINENEWLINEConjecture 7.2. Suppose that \(S(x)=\sum_{n\in\mathbf Z^d}c_n e^{inx}\) converges restrictedly rectangularly to \(0\) at every point of \(\mathbf T^d.\) Then \(S\equiv 0\).NEWLINENEWLINENEWLINETheorem 7.3. Suppose that \(S(x)=\sum_{n\in\mathbf Z^d}c_n e^{inx}\) converges spherically to \(0\) at every point of \(\mathbf T^d.\) Then \(S\equiv 0.\) NEWLINENEWLINENEWLINEThe first one is known and is associated with several names. The next two are conjectures with a good amount of empirical evidence against them. The last assertion was proved in such form by J. Bourgain. The remainder of the chapter is devoted to the detailed exposition of Bourgain's proof. Though for dimension \(2\) there exist simpler proofs and the proof in question is of real interest for \(d\geq 3\) only; the author prefers this setting to explain Bourgain's interesting and useful techniques.NEWLINENEWLINEFor the entire collection see [Zbl 0954.65001].