Further properties of an extremal set of uniqueness
David E. Grow, Matt Insall · Colloquium Mathematicum · 1998
Let T denote the group [0, 1) with addition modulo one.In [4] we presented an elementary construction of a countable, compact subset S of T which could not be expressed as the union of two H-sets, and conjectured that S is not expressible as the union of finitely many H-sets.Here we use a descriptive set theory result of S. Kahane [6] to help show that S cannot be expressed as the union of finitely many Dirichlet sets.For the connection of this problem with that of characterizing sets of uniqueness for trigonometric series on T, see [7] and [4].Let Z denote the integers and N the nonnegative integers.If x and y are real numbers then by x ≡ y we shall mean x -y ∈ Z, and in this case we identify x and y with a single point in T. A subset E of T is a set of uniqueness if the only trigonometric series ∞ n=-∞ c(n)e 2πinx on T which converges to zero for all x outside E is the zero series: c(n) = 0 for all n.A compact subset E of T is an H-set if there exists a nonempty open interval I in T such that