An upper bound for the intermediate dimensions of Bedford–McMullen carpets

The intermediate dimensions of a set \Lambda , elsewhere denoted by \dim_{\theta}\Lambda , interpolate between its Hausdorff and box dimensions using the parameter \theta\in[0,1] . For a Bedford–McMullen carpet \Lambda with distinct Hausdorff and box dimensions, we show that \dim_{\theta}\Lambda is strictly less than the box dimension of \Lambda for every \theta<1 . Moreover, the derivative of the upper bound is strictly positive at \theta=1 . This answers a question of Fraser; however, determining a precise formula for \dim_{\theta}\Lambda still remains a challenging problem.