## Publication Date:

26 September 2019

## Journal:

Forum of Mathematics, Sigma

## Last Updated:

2021-09-26T13:10:06.79+01:00

## DOI:

10.1017/fms.2019.28

## 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:

## Submitted to ORA:

Submitted

## Publication Type:

Journal Article