11 problems

Let $\gamma$ be a separated system of polynomials with total degree $D_\gamma$. For a finite set $\mathcal{X}$, let $\mathcal{M}^{\gamma}_t(\mathcal{X},[N])$ denote the associated…

Let $\gamma(T)=\big(\phi_1(T),\dots,\phi_r(T)\big)\subset\mathbb{Z}[T]$ be a separated system of $r$ polynomials with integer coefficients and total degree $D_\gamma$. For a finite…

Let $\gamma(T)=\big(\phi_1(T),\dots,\phi_r(T)\big)\subset\mathbb{Z}[T]$ be a separated system of $r$ polynomials with integer coefficients and total degree $D_\gamma$. For a finite…

Let $d\geq 1$ and let $\gamma(n)=(P_1(n),\ldots,P_d(n))$ be a polynomial curve in $\mathbb{Z}^d$, where $P_1,\ldots,P_d\in\mathbb{Z}[X]$ have separated degrees, meaning…

Höllig–Koch conjecture. Under some appropriate assumptions, a polynomial curve of degree at most $n$ can interpolate $2n$ points with optimal approximation order $2n$.

Let $\bfm{c}$ be a circular arc and let $\bfm{p}_n$ be a $G^k$ geometric approximant of degree $n$. Choose a polynomial $p_{2n,k}^*$ minimizing … where $q_{n,k}$ has degree…

Uniform coefficient conjecture. The constant $C_{P,Q}$ in the estimate

Let $k\geq 1$ be an integer and let $\mathscr{C}$ be a curve as in the paper. Write $\mathscr{H}^{k}_{\lambda}$ for the set of points satisfying the relevant Diophantine approximat…

For a smooth function $f\colon\mathbb{R}\to\mathbb{R}$, let $W_{\tau}(f)$ be the set of points defined by the simultaneous Diophantine approximation conditions used above, with inf…

Let $d\geq 3$ and let $P:{\bf R}\to{\bf R}^{d-1}$ be a polynomial of degree $N$. Write … for affine arclength measure, and let $X^P$ denote the restricted X-ray transform associate…

Equidistribution conjecture. The canonical projections onto $X$ of the dilations of $p$ equidistribute if and only if their projections onto the horizontal torus of $X$ equidistrib…