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Using the polynomial method of Dvir [On the size of Kakeya sets in finite fields.  Preprint ], we establish optimal estimates for Kakeya sets and Kakeya maximal functions associated to algebraic varieties  W  over finite fields  F . For instance, given an ( n −1)‐dimensional projective variety  W ⊂¶  n  ( F ), we establish the Kakeya maximal estimate‖sup    γ  ∍  w∑   v  ∈  γ  (  F  )|   f  (  v  )   |‖l  n   (  W  )    ≤   C   n  ,  W  ,  d|  F  |(  n  −  1  )  /  n‖  f  ‖l  n   (   F  n   )for all functions  f : F   n  → R  and  d ≥1, where for each  w ∈ W , the supremum is over all irreducible algebraic curves in  F   n   of degree at most  d  that pass through  w  but do not lie in  W , and with  C   n , W , d   depending only on  n , d  and the degree of  W ; the special case when  W  is the hyperplane at infinity in particular establishes the Kakeya maximal function conjecture in finite fields, which in turn strengthens the results of Dvir.

THE KAKEYA SET AND MAXIMAL CONJECTURES FOR ALGEBRAIC VARIETIES OVER FINITE FIELDS