Tackling thin annuli

The measure is said to have the thin annuli property (called differently by different authors), if the measure of an annuli of width $r^k$ compared to the measure of a ball of radius $r$ goes to zero (as $r$ shrinks to 0). This problematic property occurs e.g. when proving the exponential statistics of return times to shrinking balls. We will show some results and techniques useful in proving this property (at least partially).