Consider a Brownian motion in three dimensional space. We asked (TYI 37) What is the largest number of points on the path described by the motion which form an arithmetic progression? (Namely, , so that all are equal.)
Here is what you voted for
TYI37 poll: Final-results
Analysis of the poll results: Almost surely 2 is the winner with 30.14% of the 209 votes, and almost surely infinity (28.71%) comes close at second place. In the third place is almost surely 3 (14.83%), and then comes positive probability for each integer (13.4%), almost surely 5 (5.26%), almost surely 6 (2.87%), and almost surely 4 (2.39%).
Test your political intuition: which coalition is going to be formed?
Almost surely 2 (briefly AS2) and almost surely infinity (ASI) can form a government with no need for a larger coalition. But they represent two political extremes. Is AS3 politically closer to AS2 or to ASI? “k with probability p_k for every k>2” (briefly, COM) represent a complicated political massage. Is it closer to AS2 or to ASI? (See the old posts on which coalition will be formed.)
TYI37 poll: Partial results. It was exciting to see how the standing of the answers changed in the process of counting the votes.
And the correct answer is:
See the paper:
Itai Benjamini and Gady Kozma: Arithmetic progressions in the trace of Brownian motion in space
Update: See Yuval Peres’ comment with an intuitive explanation.