We prove a joint partial equidistribution result for common perpendiculars with given density on equidistributing equidistant hypersurfaces, towards a measure supported on truncated stable leaves. We recover a result of Marklof on the joint partial equidistribution of Farey fractions at a given density, and give several analogous arithmetic applications, including in Bruhat–Tits trees.

We determine, up to multiplicative constants, the number of integers $n\leq x$ that have a divisor in $(y,2y]$ and no prime factor $\leq w$. Our estimate is uniform in $x,y,w$. We apply this to determine the order of the number of distinct integers in the $N\times N$ multiplication table, which are free of prime factors $\leq w$, and the number of distinct fractions of the form $(a_{1}a_{2})/(b_{1}b_{2})$ with $1\leq a_{1}\leq b_{1}\leq N$ and $1\leq a_{2}\leq b_{2}\leq N$.

The Frobenius number $F(\boldsymbol{a})$ of a lattice point $\boldsymbol{a}$ in $\mathbb{R}^{d}$ with positive coprime coordinates, is the largest integer which can not be expressed as a non-negative integer linear combination of the coordinates of $\boldsymbol{a}$. Marklof in [The asymptotic distribution of Frobenius numbers, Invent. Math. 181 (2010), 179–207] proved the existence of the limit distribution of the Frobenius numbers, when $\boldsymbol{a}$ is taken to be random in an enlarging domain in $\mathbb{R}^{d}$. We will show that if the domain has piecewise smooth boundary, the error term for the convergence of the distribution function is at most a polynomial in the enlarging factor.