Let $f,g:\mathbb{N}\to\mathbb{T}$ be completely multiplicative functions with dense images in the complex unit circle $\mathbb{T}$. We prove that, for every non-empty open set $U \subset \mathbb{T}^2$, the set of integers $n$ such that $(f(n),g(n+1)) \in U$ has positive lower logarithmic density, unless the pair $(f,g)$ is of a special form. This result strengthens earlier theorems of Klurman and Mangerel, as well as of Charamaras, Mountakis, and Tsinas, and yields substantially simpler proofs.
We consider infinite walks in $\mathbb{N}^k$ with standard unit basis vector steps that avoid $t$ collinear points, and show that these walks exist for $(k,t) \in \{(6,3), (4,4), (3,7)\}$. In particular, our construction for $k = 3$ improves the previous bound $189$, obtained by Lidbetter, to $7$. Our results also impl...
Stijn Cambie, Erik Kalviainen, J. Shallit· 0 citations
Let $F:\mathbb{R}^3\to\mathbb{R}$ be a polynomial that is irreducible over $\mathbb{R}$ with $\text{deg} F\geq2$. We prove that, for any finite $X\subset Z(F)$ that does not concentrate on affine lines, \[ E(X)=\#\{(a,b,c,d)\in X^4: a+b=c+d\}\ll_{\text{deg} F,\,\epsilon}(\# X)^{2+\epsilon}. \] In particular, this answe...
We show that the set $\mathbb{P}-1$ of shifted primes contains infinitely many sum-product patterns of the form $\{x,x+y,xy\}$ with $x,y$ arbitrarily large distinct integers. More strongly, we can also show that, for any $k\geq 1$, the set $\mathbb{P}-1$ contains longer patterns of the form $\{x,x+y,\ldots, x+ky,xy\}$...
Let $\mathfrak{X}_k\subseteq{\sf Mat}_{m, n}$ denote the variety of $m\times n$ complex matrices with rank at most $k$. The power series and rational expressions for the Hilbert series of $\mathfrak{X}_k$ are known by geometric arguments, and equating these expressions yields a family of formulas generalizing the class...
Let $ C \subseteq \mathbb{R}^n $ be a convex set. The mapping $ f : C \longrightarrow \mathbb{R}^n $ is strictly increasing, if $ \langle f(x)-f(y), x-y \rangle>0 $ for all distinct elements $ x,y \in C $. Applying classical theorems of finite dimensional convex geometry and convex analysis, we show that the inverse fu...
We study the complete number problem for a class of self-similar spectral measures on the real line. For a spectral pair $(\mu,\Lambda)$, a real number $t$ is called complete if $t\Lambda$ is also a spectrum of $\mu$. In this paper we consider the $N$-Bernoulli convolution $\mu_{N^r,\mathcal D},\mathcal D=\{0,1,\ldots,...
