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 non-empty $E_p$ has the cardinality of the continuum. We obtain stronger conclusions for the $t$-adic Littlewood conjecture over a finite field $\mathbb F_q$. For every prime power $q$, non-emptiness of the exceptional set $E_q^{(t)}$ implies that its $1/\log(1/r)$-Hausdorff measure is infinite. Moreover, when $q$ is odd, we refine the recent construction of Lai and Sprang~\cite{LaiSprang2026} and prove that \[ \mathcal H^{h_{A_q}}(E_q^{(t)})=\infty, \] where \[ h_{A_q}(r)=\frac{1}{(\log(1/r))^{A_q}}, \qquad A_q=\frac{q-1}{2}\log_2(q-1). \] In characteristic two, the corresponding conclusion with exponent one remains conditional on the existence of a counterexample.
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...
We give an alternative proof of the $5/4$ result of Guth-Iosevich-Ou-Wang on the Falconer distance conjecture: if a compact set $E$ in the plane has Hausdorff dimension $>5/4$, then there exists $x\in E$ such that the pinned distance set $$\Delta_x(E):=\{|x-y|: y\in E\} $$ has positive Lebesgue measure. Moreover, we ar...
Let $\overline{Z}$ be a smooth projective variety with an simple normal crossing divisor $D$ such that $\Omega_{\overline{Z}}^1(\log D)$ is nef. We prove that if $Y\subset Z:=\overline Z\setminus D$ is a smooth closed subvariety with nonzero Euler characteristic, and $P$ is a perverse sheaf on $Y$ with full support, th...
Donu Arapura, Chikako Mese, D. Patel· 1 citation· ⚡1
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...
Let $X$ be the good reduction of a projective hyperk\"ahler variety of dimension $2n\geq4$. We prove that $X$ is quasi-$F$-split if and only if it is Frobenius split, equivalently if $\operatorname{H}^2_{\operatorname{crys}}(X/W)[1/p]$ has a slope-zero part. Thus its quasi-$F$-split height is $1$ or $\infty$. The proof...
Let $p \geq 5$ be a prime, let $K$ be a finite extension of $\mathbb{Q}_p$, and let $E/K$ be an elliptic curve with good reduction. Let $F_n$ denote the $n$-division polynomial of $E$. Silverman proved that if the reduction is ordinary, then for every $P \in E(K) \setminus \hat{E}(K)$ and a suitable power $q$ of $p$, t...
Yukiko Katagiri, Shinichi Kobayashi· 0 citations
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