Let $E\subset \mathbb{R}$ be a self-similar set generated by a homogeneous iterated function system $\Phi$ with contraction ratio $\rho\in (0,1)$. Assume that $\Phi$ satisfies the open set condition and $\dim_{\rm H}E<1$. Let $f$ be a $C^1$-diffeomorphism on $\mathbb{R}$. We prove that if $\log|f'(x)|/\log\rho\not\in\mathbb{Q}$ for every $x\in E\cap f^{-1}(E)$, then the upper Minkowski dimension of $f(E)\cap E$ is strictly less than the Hausdorff dimension of $E$. We also establish a quantitative dimension drop result when $E$ is a missing-digit set and $f$ is an affine map with rational slope satisfying a certain arithmetic condition. Based on these results and a result of Shmerkin [Ann. of Math., 2019], we obtain characterizations of $\gamma$ in various contexts such that $\overline{\dim}_{\rm M}\left((\gamma E+\alpha)\cap E\right)<\dim_{\rm H}E$ for every $\alpha\in\mathbb{R}$.
For $0<d\leq1/4$, let $\mathcal{C}(d)$ be the four-corner Cantor set with Hausdorff dimension $s_d=\log 4/\log(1/d)$. Peres, Simon, and Solomyak, as well as Mattila, asked for which $d$ the measure $\Hau^{s_d}(p_\theta(\mathcal{C}(d)))$ is positive for almost every direction $\theta$. It was open for the range $1/9\leq...
Let $X\subset \mathbb C^n$ be a closed pure $d$-dimensional complex analytic set. We associate to $X$ its set of projective asymptotic directions $$ \Sigma_\infty(X) :=\{\ell \in \mathbb P^{n-1}(\mathbb C):\ell\cap C_\infty(X)\ne\{0\}\}, $$ where $ C_\infty(X)$ is the total tangent cone at infinity. We prove the metric...
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 $\Sigma$ be a closed oriented surface of genus $>1$ and $M$ a complete hyperbolic 3-manifold with a marking $i:\Sigma\longrightarrow M$. We consider the case that $M$ has no parabolic cusps and at least one of the two ends is simply degenerate. For $\varGamma=\pi_1(\Sigma)$, let $\rho_M:\varGamma\longrightarrow \ma...
Let $d\geq 1$. Let $K\subset\mathbb{R}^d$ be a non-singleton self-similar set generated by a finite strongly irreducible iterated function system satisfying the open set condition, and let $\delta=\dim_{\mathrm H} K$. For $\tau>1/d$, set \[ W_d(\tau) = \left\{ \mathbf{x}\in\mathbb{R}^d: |q\mathbf{x}-\mathbf{p}|<q^{-\ta...
Let $2\le k\le n$, let $\Omega\subset\mathbb{R}^n$ be open and convex, and let $u$ be a convex viscosity solution of $\sigma_k(D^2u)=1$ in $\Omega$. We prove that the set on which $u$ fails to be locally $C^2$ has vanishing $(n-1)$-dimensional Hausdorff measure. In the intermediate range $3\le k<n$, this gives a codime...
Xi-Yu Hu· 2 citations
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