The Sidon Decomposition Problem in Abelian Groups of Bounded Torsion

Let $G$ be a compact abelian group whose dual group $\mathrm{\Gamma }=\widehat{G}$ has bounded torsion. In 1967, Malliavin-Brameret and Malliavin proved that every Sidon set in $\mathrm{\Gamma }$ is a finite union of quasi-independent sets when $\mathrm{\Gamma }$ has prime exponent. This was later extended to squarefree exponents in work of Varopoulos and Bourgain. We prove the remaining bounded-torsion case. Consequently, if $\widehat{G}$ has bounded torsion, then a subset $\mathrm{\Lambda }\subset \widehat{G}\setminus 0$ is Sidon if and only if it is a finite union of quasi-independent sets.