4

RICHARD F. BASS AND KRZYSZTOF BURDZY

of the proof is to show that there are no approximate points of increase of size

e at any level or any angle. We do this by repeated bisection of angles. Given

that there are no approximate points of increase for any angle a that is a multiple

of l/2*log(l/£), we show that there is very small probability that there is an

approximate point of increase at any level x for any angle that is a multiple of

l/2fc+1 log(l/e). We show, in fact, that these probabilities are summable in k.

The next section presents some preliminaries. Estimates for a Bessel process

of order 3 are derived in Sections 2-4. The first part of the proof is presented in

Sections 5 and 6, while Sections 7 and 8 complete the proof. Additional heuristic

arguments are given in each section.

Section 2 is devoted to a decomposition of Bessel processes of order 3. These

results are of independent interest; see Theorem 2.5, for example. We show that if

Li, Mi is a certain sequence of relative maxima and minima (see Figure 2.1), then

log(Li — Mi) has the same law as the sum of i.i.d. bilateral exponentials.

Section 3 then shows that /i-path transforms of this i.i.d. sequence satisfy

estimates akin to those of conditioned Brownian motion. Section 4 uses the results

of Sections 2 and 3 to give upper and lower bounds for the existence of approximate

points of increase for a Bessel process.

Section 5 begins the consideration of two-dimensional Brownian motion with

some estimates for exit probabilities for wedges and then continues with the two

and three angle estimates mentioned above. The second moment argument is given

in Section 6.

Section 7 has more estimates on the exit distribution of a wedge and then

proceeds with upper bounds on the probability of an approximate point of increase

in a specified direction given that there are no approximate points of increase in

nearby directions. These estimates are pulled together in Section 8, and the proofs

of Theorems 0.1 and 0.7 are completed there.

Section 9 contains a sketch of the proof of Theorem 0.6. We also include a

discussion of some open problems.

We have discussed Taylor's problem about cut lines with many people over

several years. We are grateful for their interest and advice. We would like to espe-

cially thank Davar Khoshnevisan, Robin Pemantle, Yuval Peres, Michio Shimura,

and James Taylor. We would like to thank the referee for a very careful reading of

the whole manuscript and for many valuable suggestions.