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• W3105884403 abstract "We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set $A$ whose indicator function is 'approximately multiplicative' and uniformly distributed on short intervals in a suitable sense, we show that the asymptotic density of the pattern $n+1in A$, $n+2in A$, $n+3in A$ is positive, as long as $A$ has density greater than $frac{1}{3}$. Using an inverse theorem for sumsets and some tools from ergodic theory, we also provide a theorem that deals with the critical case of $A$ having density exactly $frac{1}{3}$, below which one would need nontrivial information on the local distribution of $A$ in Bohr sets to proceed. We apply our results firstly to answer in a stronger form a question of ErdH{o}s and Pomerance on the relative orderings of the largest prime factors $P^{+}(n)$, $P^{+}(n+1), P^{+}(n+2)$ of three consecutive integers. Secondly, we show that the tuple $(omega(n+1),omega(n+2),omega(n+3)) pmod 3$ takes all the $27$ possible patterns in $(mathbb{Z}/3mathbb{Z})^3$ with positive lower density, with $omega(n)$ being the number of distinct prime divisors. We also prove a theorem concerning longer patterns $n+iin A_i$, $i=1,dots k$ in approximately multiplicative sets $A_i$ having large enough densities, generalising some results of Hildebrand on his 'stable sets conjecture'. Lastly, we consider the sign patterns of the Liouville function $lambda$ and show that there are at least $24$ patterns of length $5$ that occur with positive upper density. In all of the proofs we make extensive use of recent ideas concerning correlations of multiplicative functions." @default.
• W3105884403 created "2020-11-23" @default.
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• W3105884403 date "2019-01-01" @default.
• W3105884403 modified "2023-10-18" @default.
• W3105884403 title "VALUE PATTERNS OF MULTIPLICATIVE FUNCTIONS AND RELATED SEQUENCES" @default.
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• W3105884403 doi "https://doi.org/10.1017/fms.2019.28" @default.
• W3105884403 hasPublicationYear "2019" @default.
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