Optimally Packing a Large Square by Unit Squares
We show that a large square of sidelength $x$ can be packed by unit squares in a manner so that the wasted space $W(x) = O(x^{3/5})$.
math.AG↗
arXiv subjects
Rory McClenagan
Publications and source records attributed to Rory McClenagan.
Perfectly packing a cube by cubes of nearly harmonic sidelength
Let $d$ be an integer greater than $1$, and let $t$ be fixed such that $\frac{1}{d} < t < \frac{1}{d-1}$. We prove that for any $n_0$ chosen sufficiently large depending upon $t$, the $d$-dimensional cubes of sidelength $n^{-t}$ for $n \geq n_0$ can perfectly pack a cube of volume $\sum_{n=n_0}^\infty \frac{1}{n^{dt}}$. Our work improves upon a previously known result in the three-dimensional case for when $1/3 < t \leq 4/11 $ and $n_0 = 1$ and builds upon recent work of Terence Tao in the two-dimensional case.
math.MG↗
The Distribution of Values of $\frac{L'}{L}(1/2+ε,χ_D)$
We determine the limiting distribution of the family of values $\frac{L'}{L}(1/2+ε,χ_D)$ as $D$ varies over fundamental discriminants. Here, $0<ε<\frac12$, and $χ_D$ is the real character associated with $D$. Moreover, we also establish an upper bound for the rate of convergence of this family to its limiting distribution. As a consequence of this result, we derive an asymptotic bound for the small values of $\left|\frac{L'}{L}(1/2+ε,χ_D)\right|$.
math.NT↗