The uniform measure on a Galton-Watson tree without the XlogX condition

We consider a Galton--Watson tree with offspring distribution $\nu$ of finite mean. The uniform measure on the boundary of the tree is obtained by putting mass $1$ on each vertex of the $n$-th generation and taking the limit $n\to \infty$. In the case $E[\nu\ln(\nu)]<\infty$, this measure has been well studied, and it is known that the Hausdorff dimension of the measure is equal to $\ln(m)$ (\cite{hawkes}, \cite{lpp95}). When $E[\nu \ln(\nu)]=\infty$, we show that the dimension drops to $0$. This answers a question of Lyons, Pemantle and Peres \cite{LyPemPer97}.