Thick points for planar Brownian motion and the Erdős-Taylor conjecture on random walk

Maximal Local Time of a d-dimensional Simple Random Walk on Subsets

“…This was confirmed recently by Dembo et al [5]. In fact they proved the following more general result: Theorem C ( [5]) Let S n = n k=1 X k be an aperiodic random walk with i.i.d. increments X k ∈ Z 2 that satisfy EX 1 = 0 and E|X 1 | m < ∞ for all m < ∞.…”

“…The case d = 2 is well-known for strongly aperiodic walk (see [10], p. 75) and [5] how to weaken this condition for the aperiodic case). The case d = 1 is well-known (see e.g.…”

“…( [7] or [5]) For the simple symmetric random walk on the plane we have for any α > 0 and 0 < δ < 1, n ≥ 1…”

Arithmetic Progressions in Sets of Fractional Dimension

“…On the other hand, Assumption (A) is not expected to hold: it is known in the probabilistic literature (see e.g. [7]) that Brownian images of measures as in (62) obey an estimate similar to (A) but with an additional factor of log(ǫ −1 ), and that this is optimal. Thus our Theorem 1.2 does not apply in this case.…”

Thick points for the Cauchy process

“…Theorems 1.1 and 1.2 answer the first part of open problem (6) of that paper (also implicitly present in [9]), the second part being solved in [6]. Our work relies heavily on the techniques developed in [5] and [6], and therefore owes a substantial debt to these papers.…”