PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 72, Number 1, October 1978 LOCAL />-SIDON SETS FOR LIE GROUPS A. H. DOOLEY AND PAOLO M. SOARDI Abstract. It is shown that a compact Lie group admits no local /»-Sidon sets of unbounded degree. Let G be a compact group, and let 1 < p < 2. A subset R of the dual of G is called a local /J-Sidon set if there exists a constant B such that for every a E R and for every da X da matrix Aa, \\Aa\\p<Bdy"'\\trAaoC)\L- (1) Theorem. If G is a compact Lie group, and if R is a local p-Sidon set for G, then sup{da\a E R] < oo. Proof. We first note that, if G is an arbitrary compact group, R is a />-Sidon set for G, and if r > 1, then there exists a constant <cr such that for all a E R IIX.II, < Krd2Jp' (2) where x„(x) = tr(o(x)). To see this, we first use a simple duality argument to see that (1) is equivalent to: there exists a constant C such that for every a E R and for every d„ X da matrix A„, there exists g G L\G) such that g(o) = Aa, and ||g||i < Cdyp'\\Aa\\ ,. Thus for every a E R and for every da X d0 unitary matrix W, there exists gw E L\G) with gw = W*, and \\gw\\x < Cdal/P'\\ W*\\p, = d2/"'. Since x, = gw * (tr(W■ a(-))) we have IIXJI,< \\gwh[fGHW-a(x))\rdx^ <Cd2/"^fG\tr(W-a(x))\rdx)j . Hence, integrating over the da x da unitary group, Gbl(d0)with respect to normalized Haar measure dW, and using Holder's inequality, we obtain HXJ, < Cd2/"' If f \tr(W- o(x))\r dWdx) M' = Cd2/p\f \tr W\rdw) VIM) I Received by the editors February 7, 1978. AMS (MOS) subjectclassifications (1970). Primary 42A44,43A14. © American Mathematical Society 1978 125 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use