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With mathematical jargon:
No distinct same-sized copies of self overlap on a subset with positive measure in the Hausdorff measure using the Hausdorff dimension.
For a covering of a fractal by finitely many scaled down copies of itself, the condition of that no two have an intersection with positive measure is equivalent to the condition that the Hausdorff dimension coincides with the similarity dimension.
(There is another similar condition in this context called the "open set condition" which implies this but is not equivalent. The open set condition is equivalent to the condition that the Hausdorff measure using the similarity dimension is nonzero.) | 
