Let $f:\mathbb{Z}\to\mathbb{C}$ be a multiplicative function. Assume there exist integers $a>1$ and $d>1$ such that $(a,d)=1$ and let $\mathcal{P}_{a,d}=\{a+kd:k\in\mathbb{Z}\}$. Under mild extra conditions on $d$ and $f(a)$, we prove that $f(n)=n\chi(n)$ for all $n$ outside an explicit exceptional set depending on $d$, and some Dirichlet character $\chi$.
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)...
Fix $s\in\mathbb{N}$. We prove that there exists a subadditive density on $\mathbb{N}$ such that, for every polynomial $P\in\mathbb{Z}[y]$ with $P(0)=0$, every set of positive density contains configurations $\{x,x+P(y),xy^s\}$ for arbitrarily large $x>y\geq 2$. When $s=1$, this strengthens Moreira's partition-regulari...
Let $\mu$ be a log-concave probability measure on $\mathbb R^n$ and let $f\colon\mathbb R^n\to\mathbb R^k$ be a polynomial mapping of degree at most $d$. We show that \[ \mu(f\in A) \le C\bigl(\lambda_k(A)\bigr)^{\frac{1}{k(d-1)+1}} \] for every Borel set $A\subset\mathbb R^k$ whenever the image measure $\mu\circ f^{-1...
Let $k=\overline{\mathbb{F}_2}$. We construct a nine-dimensional $F$-rational affine variety $X$ admitting a locally closed embedding $X\hookrightarrow \mathbb{P}_k^{19}$ together with a dense open set $U\subseteq (\mathbb{P}^{19})^\vee$ such that $X\cap H$ is not $F$-rational (or even $F$-injective) for all $H\in U$....
Let $N_n^{\mathrm{tor}}(X)$ be the number of isomorphism classes of $n$-dimensional algebraic tori over $\mathbb{Q}$ whose Artin conductor is bounded by $X$. We prove that there is an absolute constant $C>0$ such that, for every positive integer $n \ge 2$, $N_n^{\mathrm{tor}}(X)\ll_n X^{\exp(C(\log n)^2)}$. The proofs...
Let $f$ be a regular endomorphism of $\mathbb{A}_{\mathbb{C}}^N$ and let $C\subseteq\mathbb{A}_{\mathbb{C}}^N$ be an irreducible curve. Suppose $C$ has an infinite intersection with the $f$-orbit of a point $x\in\mathbb{A}^N(\mathbb{C})$. Then the normalization of $C$ is isomorphic to either $\mathbb{A}^1$ or $\mathbb{...
She Yang, A. Zheng· 0 citations
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