correlations of bounded multiplicative functions (as developed recently in [@jt]) to obtain new results about the distribution of sign patterns $(f_1(n+1), f_2(n+2),\dots, f_k(n+k))$ of various bounded multiplicative functions $f_1,\dots,f_k$ such as the Liouville function $\lambda(n)$, as well as generalisations such as $e^{2\pi i \Omega(n)/m}$ for a fixed natural number $m$, where $\Omega(n)$ denotes the number of prime factors of $n$ (counting multiplicity).

Results analogous to Theorem \[inv-1\] are known when the connected group $G$ is replaced by the discrete group ${\mathbb{Z}}/p{\mathbb{Z}}$: see [@freiman], [@rodseth], [@serra], [@rect], [@blr], [@g], as well as some further discussion in [@hgz]. In the recent paper [@candela], these results (particularly those in [@g]) are used to establish the $G={\mathbb{R}}/{\mathbb{Z}}$ case of Theorem \[inv-1\]. On the integers ${\mathbb{Z}}$, a version of Theorem \[inv-2\] when $A, B \subset {\mathbb{Z}}$ have the same cardinality was obtained very recently in [@shao Corollary 5.2].

Acknowledgments
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The author was supported by a Simons Investigator grant, the James and Carol Collins Chair, the Mathematical Analysis & Application Research Fund Endowment, and by NSF grant DMS-1266164. The author is indebted to Joni Teräväinen for key discussions that led to the author pursuing this question, and for helpful comments and corrections, and to Ben 