3-systems which do not represent a real number and its square
Anthony Poëls · arXiv (Cornell University) · 2018
In 2013 Schmidt and Summerer showed that the parametric successive minima function $\mathbf{L}_\mathbf{u}$ of a given vector $\mathbf{u}\in\mathbb{R}$ can be approximated up to a bounded difference by a function from a certain class. Roy recently proved that the same is true within a smaller class of functions called $n$-systems. Conversely, given an $n$-system, Roy also showed that there exists a point $\mathbf{u}\in\mathbb{R}$ whose associated function $\mathbf{L}_\mathbf{u}$ is approximated by this $n$-system up to a bounded difference. In this paper we study the case $n = 3$ and we construct $3$-systems such that there is no vector $\mathbf{u}$ of the form $(1,\xi,\xi^2)$ whose associated function $\mathbf{L}_\mathbf{u}$ may be approximated by these $n$-systems up a to a bounded difference.