Lipschitz functions and spectral synthesis

Abstract:

An $S$-set in the circle group $T$ is a closed subset $S$ of $T$ for which $\overline {j(S)} = k(S)$. We construct a non-$S$-set $S$ satisfying \[ \bigcup \limits _{\alpha > 0} {{{\operatorname {Lip} }_\alpha }(T) \cap k(S) \subset \overline {j(S)} .} \] Thus ${\operatorname {Lip} _\alpha }(T) \cap A(T)$ is not a big enough part of $A(T)$ to test the synthesizability of a given closed subset of $T$.