3 problems

Let $\alpha$ be real, and let $a,b$ be coprime integers with $b\to\infty$ and $a/b\to\alpha$. Define $c_\alpha$ to be the variational quantity given in the source by the infimum of…

Let $R$ be a finite family of slopes containing $0,1,\infty$, of cardinality $k+3$, and let $s$ be a slope not in $R$. Let $D(R;s)$ denote the rational complexity appearing in the…

Let $R$ range over finite sets of rational slopes, let $s$ be an output slope, and let ${\operatorname{SD}}(R;s)$ denote the associated sum-difference exponent. Arithmetic Kakeya c…