Lacunarity for Compact Groups

Abstract

Let G be a compact Abelian group with character group X. A subset Δ of X is called a [unk]_q set (1 < q < ∞) if for all trigonometric polynomials f = [unk]_k=1 ⁿ α_kχ_k (χ₁,...,χ_n [unk] Δ) an inequality ∥f∥_q [unk] [unk]∥f∥₁ obtains, where [unk] is a positive constant depending only on Δ. The subset Δ is called a Sidon set if every bounded function on Δ can be matched by a Fourier-Stieltjes transform. It is known that every Sidon set is a [unk]_q set for all q. For G = T, X = Z, Rudin (J. Math. Mech., 9, 203 (1960)) has found a set that is [unk]_q for all q but not Sidon. We extend this result to all infinite compact Abelian groups G: the character group X contains a subset Δ that is [unk]_q for all q, 1 < q < ∞, but Δ is not a Sidon set.