Free Actions on Products of Spheres at High Dimensions

A classical conjecture in transformation group theory states that if $G=(\text{bbZ}\mathrm{/}p{)}^r$ acts freely on a product of $k$ spheres $S^{n_1} \times … \times S^{n_k}$,then$r\leq k$. We prove this conjecture in the case where the average of the dimensions $n_i$ is large compared to the differences $\mathrm{\mid }{n}_i-n_j\mathrm{\mid }$ between the dimensions.