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In a recent paper of Ellenberg, Oberlin, and Tao [The Kakeya set and maximal conjectures for algebraic varieties over finite fields.  Mathematika   56  (2010), 1–25], the authors asked whether there are Besicovitch phenomena inF  q[  [  t  ]  ]   n    . In this paper, we answer their question in the affirmative by explicitly constructing a Kakeya set of measure zero inF  q[  [  t  ]  ]   n    . Furthermore, we prove that any Kakeya set inF  q[  [  t  ]  ]   2or    Z   p   2    is of Minkowski dimension 2.

KAKEYA SETS OVER NON‐ARCHIMEDEAN LOCAL RINGS