Recurrence for vertex-reinforced random walks on Z with weak reinforcements

We prove that any vertex-reinforced random walk on the integer lattice with non-decreasing reinforcement sequence $w$ satisfying $w(k)=o(k^{\alpha })$ for some $\alpha <1\mathrm{/}2$ is recurrent. This improves on previous results of Volkov (2006) and Schapira (2012).