Arithmetic progressions in space!

Last time I introduced a neat little question: Is it true that every infinite path in contains arbitrarily long arithmetic progressions? For example, in the following path, the squares colored in red have coordinates , which form an arithmetic progression of length 3, where every two consecutive points differ by . As it turns out, … More Arithmetic progressions in space!

There and back again, part 3

This is part three in a three-part series about recurrence and transience on the integer lattices. Here is part one. Here is part two. Weary and tired, we are approaching the end of our coin-flipping, lattice-exploring adventures, but with one aching question still burning deep within our hearts: Is the random walk on the three-dimensional … More There and back again, part 3