Joni Teräväinen and I have just uploaded to the arXiv our paper “Value patterns of multiplicative functions and related sequences“, submitted to Forum of Mathematics, Sigma. This paper explores how to use recent technology on correlations of multiplicative (or nearly multiplicative functions), such as the “entropy decrement method”, in conjunction with techniques from additive combinatorics, to establish new results on the sign patterns of functions such as the Liouville function ${\lambda}$. For instance, with regards to length 5 sign patterns $\displaystyle (\lambda(n+1),\dots,\lambda(n+5)) \in \{-1,+1\}^5$

of the Liouville function, we can now show that at least ${24}$ of the ${32}$ possible sign patterns in ${\{-1,+1\}^5}$ occur with positive upper density. (Conjecturally, all of them do so, and this is known for all shorter sign patterns, but unfortunately ${24}$ seems to be the limitation of our methods.)

The Liouville function can be written as ${\lambda(n) = e^{2\pi i \Omega(n)/2}}$, where ${\Omega(n)}$ is the number of prime factors of ${n}$ (counting multiplicity). One can also consider the variant ${\lambda_3(n) = e^{2\pi i \Omega(n)/3}}$, which is a completely multiplicative function taking values in the cube roots of unity ${\{1, \omega, \omega^2\}}$. Here we are able to show that all ${27}$ sign patterns in ${\{1,\omega,\omega^2\}}$ occur with positive lower density as sign patterns ${(\lambda_3(n+1), \lambda_3(n+2), \lambda_3(n+3))}$ of this function. The analogous result for ${\lambda}$ was already known (see this paper of Matomäki, Radziwiłł, and myself), and in that case it is even known that all sign patterns occur with equal logarithmic density ${1/8}$ (from this paper of myself and Teräväinen), but these techniques barely fail to handle the ${\lambda_3}$ case by itself (largely because the “parity” arguments used in the case of the Liouville function no longer control three-point correlations in the ${\lambda_3}$ case) and an additional additive combinatorial tool is needed. After applying existing technology (such as entropy decrement methods), the problem roughly speaking reduces to locating patterns ${a \in A_1, a+r \in A_2, a+2r \in A_3}$ for a certain partition ${G = A_1 \cup A_2 \cup A_3}$ of a compact abelian group ${G}$ (think for instance of the unit circle ${G={\bf R}/{\bf Z}}$, although the general case is a bit more complicated, in particular if ${G}$ is disconnected then there is a certain “coprimality” constraint on ${r}$, also we can allow the ${A_1,A_2,A_3}$ to be replaced by any ${A_{c_1}, A_{c_2}, A_{c_3}}$ with ${c_1+c_2+c_3}$ divisible by ${3}$), with each of the ${A_i}$ having measure ${1/3}$. An inequality of Kneser just barely fails to guarantee the existence of such patterns, but by using an inverse theorem for Kneser’s inequality in this previous paper of mine we are able to identify precisely the obstruction for this method to work, and rule it out by an ad hoc method.

The same techniques turn out to also make progress on some conjectures of Erdös-Pomerance and Hildebrand regarding patterns of the largest prime factor ${P^+(n)}$ of a natural number ${n}$. For instance, we improve results of Erdös-Pomerance and of Balog demonstrating that the inequalities $\displaystyle P^+(n+1) < P^+(n+2) < P^+(n+3)$

and $\displaystyle P^+(n+1) > P^+(n+2) > P^+(n+3)$

each hold for infinitely many ${n}$, by demonstrating the stronger claims that the inequalities $\displaystyle P^+(n+1) < P^+(n+2) < P^+(n+3) > P^+(n+4)$

and $\displaystyle P^+(n+1) > P^+(n+2) > P^+(n+3) < P^+(n+4)$

each hold for a set of ${n}$ of positive lower density. As a variant, we also show that we can find a positive density set of ${n}$ for which $\displaystyle P^+(n+1), P^+(n+2), P^+(n+3) > n^\gamma$

for any fixed ${\gamma < e^{-1/3} = 0.7165\dots}$ (this improves on a previous result of Hildebrand with ${e^{-1/3}}$ replaced by ${e^{-1/2} = 0.6065\dots}$. A number of other results of this type are also obtained in this paper.

In order to obtain these sorts of results, one needs to extend the entropy decrement technology from the setting of multiplicative functions to that of what we call “weakly stable sets” – sets ${A}$ which have some multiplicative structure, in the sense that (roughly speaking) there is a set ${B}$ such that for all small primes ${p}$, the statements ${n \in A}$ and ${pn \in B}$ are roughly equivalent to each other. For instance, if ${A}$ is a level set ${A = \{ n: \omega(n) = 0 \hbox{ mod } 3 \}}$, one would take ${B = \{ n: \omega(n) = 1 \hbox{ mod } 3 \}}$; if instead ${A}$ is a set of the form ${\{ n: P^+(n) \geq n^\gamma\}}$, then one can take ${B=A}$. When one has such a situation, then very roughly speaking, the entropy decrement argument then allows one to estimate a one-parameter correlation such as $\displaystyle {\bf E}_n 1_A(n+1) 1_A(n+2) 1_A(n+3)$

with a two-parameter correlation such as $\displaystyle {\bf E}_n {\bf E}_p 1_B(n+p) 1_B(n+2p) 1_B(n+3p)$

(where we will be deliberately vague as to how we are averaging over ${n}$ and ${p}$), and then the use of the “linear equations in primes” technology of Ben Green, Tamar Ziegler, and myself then allows one to replace this average in turn by something like $\displaystyle {\bf E}_n {\bf E}_r 1_B(n+r) 1_B(n+2r) 1_B(n+3r)$

where ${r}$ is constrained to be not divisible by small primes but is otherwise quite arbitrary. This latter average can then be attacked by tools from additive combinatorics, such as translation to a continuous group model (using for instance the Furstenberg correspondence principle) followed by tools such as Kneser’s inequality (or inverse theorems to that inequality).