identify rectangles $T$ with large $\mu$ measure and call them bad rectangles. Then for each bad rectangle $T$, we will identify the part of $\mu$ with physical support in $T$ and frequency support in the direction of $T$, and remove that part. Here is the precise definition.

We consider a sequence of scales $R_0$, $R_1$, $R_2$, etc. Here $R_0$ is a large number that we will choose later and $R_j = 2^j R_0$. Cover the annulus $R_{j-1} \le |\omega| \le R_j$ by rectangular blocks $\tau$ with dimensions approximately $R_j^{1/2} \times R_j$. The long direction of each block $\tau$ is the radial direction. We choose a partition of unity subordinate to this cover, so that

$$1 = \psi_0 + \sum_{j \ge 1, \tau} \psi_{j, \tau}.$$

Let $\delta > 0$ be a small constant.

For each $(j, \tau)$, cover the unit disk with tubes $T$ of dimensions approximately $R_j^{-1/2 + \delta} \times 1$ with the long axis parallel to the long axis of $\tau$. Let ${\mathbb{T}}_{j, \tau}$ be the collection of all these tubes, and let $\eta_T$ be a partition of unity subordinate to this covering, so that for each choice of $j$ and $\tau$, $ \sum_{T \in {\mathbb{T}}_{j, \tau}} \eta_T $ 