论文标题
在乘法整数的作用下，“自我伴随海绵”的尺寸不变
Dimensions of "self-affine sponges" invariant under the action of multiplicative integers
论文作者
论文摘要
令$ m_1 \ geq m_2 \ geq 2 $为整数。我们考虑产品符号序列空间的子集$（\ {0，\ cdots，m_1-1 \} \ times \ {0，\ cdots，m_2-1 \}）^{\ Mathbb {n}^*} $，它们是在倍增整体的半群落下是不变的。这些集合是在肯尼，佩雷斯和Solomyak之后定义的，并使用固定的整数$ Q \ geq 2 $。我们计算这些集合投影的Hausdorff和Minkowski尺寸在单位正方形的仿射网格上。我们的Hausdorff尺寸公式的证明是通过一类研究集合的某些Borel概率度量进行的。这扩展了自动式塞尔平塞斯基地毯的众所周知的结果。但是，我们在证明中使用的组合论点比自相似的情况更精致，并涉及一个新的参数，即$ j = \ left \ lfloor \ log_q \ left（\ frac {\ frac {\ log（m_1）} {\ log log log log log（m_2）} \ right）然后，我们将结果概括为在尺寸$ d \ geq 2 $中定义的相同子集。在那里，情况更加微妙，我们的公式涉及$ 2D-3 $参数的集合。
Let $m_1 \geq m_2 \geq 2$ be integers. We consider subsets of the product symbolic sequence space $(\{0,\cdots,m_1-1\} \times \{0,\cdots,m_2-1\})^{\mathbb{N}^*}$ that are invariant under the action of the semigroup of multiplicative integers. These sets are defined following Kenyon, Peres and Solomyak and using a fixed integer $q \geq 2$. We compute the Hausdorff and Minkowski dimensions of the projection of these sets onto an affine grid of the unit square. The proof of our Hausdorff dimension formula proceeds via a variational principle over some class of Borel probability measures on the studied sets. This extends well-known results on self-affine Sierpinski carpets. However, the combinatoric arguments we use in our proofs are more elaborate than in the self-similar case and involve a new parameter, namely $j = \left\lfloor \log_q \left( \frac{\log(m_1)}{\log(m_2)} \right) \right\rfloor$. We then generalize our results to the same subsets defined in dimension $d \geq 2$. There, the situation is even more delicate and our formulas involve a collection of $2d-3$ parameters.
