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We study the existence of various sign and value patterns in sequencesdefined by multiplicative functions or related objects. For any set $A$ whoseindicator function is 'approximately multiplicative' and uniformly distributedon short intervals in a suitable sense, we show that the asymptotic density ofthe pattern $n+1\in A$, $n+2\in A$, $n+3\in A$ is positive, as long as $A$ hasdensity greater than $\frac{1}{3}$. Using an inverse theorem for sumsets andsome tools from ergodic theory, we also provide a theorem that deals with thecritical case of $A$ having density exactly $\frac{1}{3}$, below which onewould need nontrivial information on the local distribution of $A$ in Bohr setsto proceed. We apply our results firstly to answer in a stronger form aquestion of Erd\H{o}s and Pomerance on the relative orderings of the largestprime factors $P^{+}(n)$, $P^{+}(n+1), P^{+}(n+2)$ of three consecutiveintegers. Secondly, we show that the tuple$(\omega(n+1),\omega(n+2),\omega(n+3)) \pmod 3$ takes all the $27$ possiblepatterns in $(\mathbb{Z}/3\mathbb{Z})^3$ with positive lower density, with$\omega(n)$ being the number of distinct prime divisors. We also prove atheorem concerning longer patterns $n+i\in A_i$, $i=1,\dots k$ in approximatelymultiplicative sets $A_i$ having large enough densities, generalising someresults of Hildebrand on his 'stable sets conjecture'. Lastly, we consider thesign patterns of the Liouville function $\lambda$ and show that there are atleast $24$ patterns of length $5$ that occur with positive upper density. Inall of the proofs we make extensive use of recent ideas concerning correlationsof multiplicative functions.

Value patterns of multiplicative functions and related sequences