Файл:Assouad dimension.svgThe Assouad dimension of the Sierpiński triangle is equal to its Hausdorff dimension, <math>\alpha = \frac{\log(3)}{\log(2)}</math>. In the illustration, we see that for a particular choice of Шаблон:Mvar, Шаблон:Mvar, and Шаблон:Mvar, <math display="block"> N_{r}(B_{R}(x) \cap E) = 3 = 2^\alpha = \left( \frac{R}{r} \right)^{\alpha}.</math> For other choices, the constant Шаблон:Mvar may be greater than 1, but is still bounded.
Let <math>(X, d)</math> be a metric space, and let Шаблон:Mvar be a non-empty subset of Шаблон:Mvar. For Шаблон:Math, let <math>N_{r}(E)</math> denote the least number of metric open balls of radius less than or equal to Шаблон:Mvar with which it is possible to cover the set Шаблон:Mvar. The Assouad dimension of Шаблон:Mvar is defined to be the infimal <math>\alpha \ge 0</math> for which there exist positive constants Шаблон:Mvar and <math>\rho</math> so that, whenever
<math display="block">0 < r < R \leq \rho,</math>
the following bound holds:
<math display="block">\sup_{x \in E} N_{r}(B_{R}(x) \cap E) \leq C \left( \frac{R}{r} \right)^{\alpha}.</math>
The intuition underlying this definition is that, for a set Шаблон:Mvar with "ordinary" integer dimension Шаблон:Mvar, the number of small balls of radius Шаблон:Mvar needed to cover the intersection of a larger ball of radius Шаблон:Mvar with Шаблон:Mvar will scale like Шаблон:Math.
The Assouad dimension of a metric space is always greater than or equal to its upper box dimension, which in turn is greater than or equal to the Hausdorff dimension.[4]
