Set of uniqueness

Remarks

Sets of uniqueness are often called U-sets, and sets of multiplicity M-sets. A subset of a set of uniqueness is again a set of uniqueness.

Example: the empty set is a set of uniqueness; this is Cantor's uniqueness theorem. Every countable set is also a set of uniqueness, a result of W. H. Young.

Non-example: let $E$ contain an open arc $I$ of $\mathbb{T}$. Choose a nonzero infinitely differentiable function $g$ on $\mathbb{T}$ that vanishes outside $I$. Its Fourier series converges absolutely to $g$ at every point, so it converges to $0$ outside $E$. Its coefficients are not all $0$, because $g \ne 0$, so $E$ is a set of multiplicity.

More generally every set of positive Lebesgue measure is a set of multiplicity, and a Borel set of uniqueness has measure zero. The converse fails: Menshov's example is a closed set of measure zero that is a set of multiplicity. The middle-thirds Cantor set is a set of uniqueness.