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Redmond McNamara

Publications and source records attributed to Redmond McNamara.

3 recordsLinked to original sources
arXiv2023

A counterexample to Hildebrand's conjecture on stable sets

We provide a counterexample to a conjecture of Hildebrand which states that if $\S$ has positive lower density and is stable i.e. for all $d$, $n$ is in $\mathcal{S}$ if and only if $dn$ is in $\mathcal{S}$ except on a set of density $0$ then $\mathcal{S} \cap (\mathcal{S}+1) \cap (\mathcal{S}+2)$ has positive lower density and in particular is nonempty. We further show there exists a stable set of density $1 -\frac{1}{q-1}$ such that $\mathcal{S} \cap \cdots \cap (\mathcal{S} + q -1) = \emptyset$ when $q$ is a prime, matching a bound proven by Hildebrand. Finally, we construct a function $f : \mathbb{N} \rightarrow \{\pm 1\}$ such that $f(pn) = -f(n)$ for all but a $0$ density set of $n$ depending on the prime $p$ but which fails the analogues of Sarnak and Chowla's conjectures.

math.CO↗
arXiv2020

Sarnak's Conjecture for Sequences of Almost Quadratic Word Growth

We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce the main theorem from a variant which bounds the correlations between multiplicative functions and sequences with subquadratically many sign patterns which occur with positive logarithmic density. This allows us to actually prove that our multiplicative functions do not locally correlate with sequences of subquadratic word growth. We also prove a conditional result which shows that if the $κ-1$-Fourier uniformity conjecture holds then the Liouville function does not correlate with sequences with $O(n^{t-\varepsilon})$ many words of length $n$ where $t = κ(κ+1)/2$. We prove a variant of the $1$-Fourier uniformity conjecture where the frequencies are restricted to any set of box dimension $< 1$.

math.DS↗