Counterexamples to Wang's Conjecture

In this paper, we give a negative answer to Wang's conjecture. For every $n\geq 3$, there exist $\varepsilon>0$ and $δ\in (0,1)$ such that, for all $q$ and $λ$ satisfying \[ \frac{n}{n-2}-\varepsilon<q\le \frac{n}{n-2}, \qquad \frac{1}{q-1}\cdot δ<λ\le \frac{1}{q-1}, \] there exists a bounded Euclid