https://en.wikipedia.org/wiki/Hausdorff_dimension

https://en.wikipedia.org/wiki/Open_set_condition

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.)

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 none overlapping on a subset 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.)

In mathematical jargon:

No distinct smaller 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 none overlapping on a subset 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.)

In mathematical jargon:

No distinct smaller 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 distinct scaled down copies of itself, this 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.)

In mathematical jargon:

No distinct smaller copies of self overlap on a subset with positive measure in the Hausdorff measure using the Hausdorff dimension.

For fractals covered by finitely many distinct scaled down copies of themselves, this is equivalent to the condition that the Hausdorff dimension coincides with the similarity dimension.

(There is another similar condition 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.)

https://en.wikipedia.org/wiki/Open_set_condition

No same-sized copies of self overlap vs. distinct same-sized copies overlap.

No same-sized copies of self (appearing within self) overlap vs. distinct same-sized copies overlap.