Journal title
Forum of Mathematics, Sigma
DOI
10.1017/fms.2019.28
Last updated
2021-10-28T05:10:42.067+01:00
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 $n+1\in A$, $n+2\in A$, $n+3\in A$ 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 Erd\H{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 $(\mathbb{Z}/3\mathbb{Z})^3$ with positive lower density, with
$\omega(n)$ being the number of distinct prime divisors. We also prove a
theorem concerning longer patterns $n+i\in A_i$, $i=1,\dots k$ 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 density. In all of
the proofs we make extensive use of recent ideas concerning correlations of
multiplicative functions.
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 $n+1\in A$, $n+2\in A$, $n+3\in A$ 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 Erd\H{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 $(\mathbb{Z}/3\mathbb{Z})^3$ with positive lower density, with
$\omega(n)$ being the number of distinct prime divisors. We also prove a
theorem concerning longer patterns $n+i\in A_i$, $i=1,\dots k$ 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 density. In all of
the proofs we make extensive use of recent ideas concerning correlations of
multiplicative functions.
Symplectic ID
991706
Download URL
http://arxiv.org/abs/1904.05096v1
Submitted to ORA
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Publication type
Journal Article
Publication date
26 September 2019