# For each $\alpha > 2$ there is an Infinite Binary Word with Critical Exponent $\alpha$

### Abstract

The critical exponent of an infinite word ${\bf w}$ is the supremum of all rational numbers $\alpha$ such that ${\bf w}$ contains an $\alpha$-power. We resolve an open question of Krieger and Shallit by showing that for each $\alpha > 2$ there is an infinite binary word with critical exponent $\alpha$.