Periodic distributions and non-Liouville numbers

Abstract

In an earlier paper (Translation-invariant linear forms and a formula for the Dirac measure, J. Functional Analysis 8 (1971), 173–188) the author proved that the ratio α β of two real numbers is a non-Liouville number if and only if there exist two Schwartz distributions A and B with compact supports on the real line such that δ′ = A − τ α A + B − τ β B. The present paper presents (in Section 3) a completely new and more elementary proof of this result (stated fully as Theorems 1 and 2 at the end of Section 1) based on some fundamental properties of the mapping H 0 = − Δ and H = H 0 + T, where T which are established in Section 2. Further connections with Diophantine approximation (badly approximable numbers and Roth's Theorem) are presented in Section 4 where it is proved that the orders of the distributions A and B are always ⩾2 (Theorem 3) and almost always =3 (Theorem 4). Section 5 contains some partial results (Theorem 5 and Corollary 2 of Theorem 6) on the analogous question (as yet unsettled) for the Banach space C(T) of all continuous functions on the circle group T, and connects this problem with Sidon sets of integers.