AN EXTREMAL SET OF UNIQUENESS? BY

Let T denote the group [0, 1) with addition modulo one, let Z denote the integers, and let E be a subset of T. E is a set of uniqueness if the only trigonometric series n=− ∞ c(n)e 2piinx on T which converges to zero for all x outside E is the zero series; E is an H-set if there exists a nonempty open interval I in T such that N(E; I) = {n ∈ Z | nx 6 ∈ I for all x ∈ E} is infinite; E is a Dirichlet set if N(E; (ε, 1 − ε)) is infinite for all ε> 0. Let E(0) = E and inductively define E(n) as the set of limit points of E(n−1). If there exists a positive integer n such that E(n) is empty then E has finite (Cantor–Bendixson) rank; in this case, the least such integer n is the rank of E. Every finite subset of T is a Dirichlet set [3], every Dirichlet set is clearly an H-set, and every H-set is a set of uniqueness [5]. Cantor [2] showed that any set of finite rank is a set of uniqueness, and a similar argument shows that every countable closed set E in T is a set of uniqueness [4, p. 32]. By a result of W. H. Young [7], the hypothesis that E is closed can be deleted without changing the conclusion. For an introduction to the vast literature on sets of uniqueness see [1], [8], and [4]. The purpose of this note is an elementary construction of a closed set S of rational numbers in T which necessarily is a set of uniqueness, but which cannot be expressed as the union of two H-sets. We conjecture, moreover, that S is not the union of a finite number of H-sets. In this case, S would be extremal among the closed subsets of T which are expressible as a countable union of H-sets. (By a nonconstructive argument [4, pp. 127–128], it is known that possibly uncountable extremal sets of this type exist.) The extremality of S, consequently, would provide insight into the long-standing problem of characterizing the closed sets of uniqueness in T. Given x in T, let x = k=0 xk2 −k, xk ∈ {0, 1}, denote its binary ex-pansion, and write x = x0.x1x2x3...; this expression for x is unique if the terminating expansion is chosen whenever possible. Let S−1 = {0} and, for each nonnegative integer n, let Sn signify the set of all x = x0.x1x2x3... in T such that j=0 xj = n+ 1 and xj = 0 if 0 ≤ j ≤ n