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 $1/9\leq d\leq1/6$. In this paper, we show that, for $\delta<d\leq1/6$, there is a set $\IP(d)$ of positive Lebesgue measure such that for almost every $\theta\in\IP(d)$, $\Hau^{s_d}(p_\theta(\mathcal{C}(d)))=0$, where $\delta=0.155124983896014\ldots$ is the unique zero in $(1/9,1/6)$ of the polynomial $P(d)=1-7d+3d^2+4d^3-2d^4-d^5$.
Let $A\subset\mathbb{R}^d$, $d\geq3$, be the attractor of a self-similar IFS whose rotations generate a dense subgroup of $\mathrm{SO}(d)$. For every $1\leq k<\dim_{\mathrm{H}} A$ and $V\in\mathrm{Gr}(d,k)$, we prove that $\pi_VA$ has positive $k$-dimensional Lebesgue measure uniformly in $V$, and $\dim_{\mathrm{H}}(A\...
Let $d\geq 2$, $0<\alpha<1$, and let $\Gamma_k$ be the successive arrival times of a standard Poisson process on $(0,\infty)$. Given independent uniform directions $\varepsilon_k\in S^{d-1}$, independent of $(\Gamma_k)$, we consider the random countable stable zonotope $Z_\alpha=\bigoplus_{k=1}^{\infty}\Gamma_k^{-1/\al...
The uniform Littlewood conjecture (ULC), introduced by Bandi, Fregoli and Kleinbock, asserts in the two-number case that $$ \lim_{Q\to\infty} Q\min_{1\le n\le Q}\|n\xi\|\,\|n\zeta\|=0 $$ for all real $\xi,\zeta$. It is proven to hold for almost every pair $(\xi,\zeta)$. Schleischitz, however, has recently disproved the...
We establish the Hausdorff dimension for exactly approximable matrices for all dimensions, including the inhomogeneous case. More precisely, let $\psi:\N\to(0,\infty)$ be a non-increasing approximation function and let $\tau_\psi$ denote its lower order. For $m,n\ge1$ and $\boldsymbol\theta\in\mathbb R^m$, denote $\Exa...
We prove that if the exceptional set $E_p$ for the $p$-adic Littlewood conjecture is non-empty, then its logarithmic Hausdorff dimension is at least one. More precisely, whenever $E_p$ is non-empty, it has positive Hausdorff measure with respect to the gauge function $ h(r)=\frac{1}{\log(1/r)}. $ In particular, every n...
For a $1$-rectifiable set $E \subset \mathbb{R}^d$ of positive length, a theorem of Federer shows that, among any $d$ linearly independent orthogonal projections of $E$, at least one has positive length. We develop a version of this finite-witness principle for generalized curve projections. Given scalar-valued mapping...
Caleb Marshall· 1 citation
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