Last year I read a nice blog articles Recent progress on the Kakeya conjecture and have several questions with this article.
follows the proof strategy called Multiscale analysis,although we can use the estimate with large $ \delta_1$ to get estimate with small $ \delta_2$,(may be loss some $ \delta^c$ in the inequality in this way),but the main difficult is we should proof the new tubes with scales $ \delta_2$ is contains in the the olders.as soon as we proof this ,to obtain a lower bound of minkwoski dimension with kakeya set, suffice to get following estimate :
the new cubes with scale $ \delta_2$ contains a positive constants volumes of every old cubes with scale $ \delta_1$.
this type of estimate is easy to attain because it is very similar to the “principle of close packing of spheres”.
in general ,we should not expect this claims:
the new tubes with scales $ \delta_2$ is contains in the the older.
but if we can proof in some sense most of new tubes comes from this way maybe we can make progress on the original problem.
roughly speaking,we should partition the whole set of $ T_{\delta}$ into two part,comes from old ones or not,for the first kind i.e contains in a old one,use the way explained above to treat.the second kind we need to proof the influence is very some or we can sometimes use the cubes from the first kind to instead the cubes from second kind and the measure of $ |A|_{\delta}$ change little.