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- W3100507412 abstract "Using the polynomial method of Dvir cite{dvir}, 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 $W subset P^n(F)$, we establish the Kakeya maximal estimate $$ | sup_{gamma ni w} sum_{v in gamma(F)} |f(v)| |_{ell^n(W)} leq C_{n,W,d} |F|^{(n-1)/n} |f|_{ell^n(F^n)}$$ for all functions $f: F^n to R$ and $d geq 1$, where for each $w in W$, 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." @default.
- W3100507412 created "2020-11-23" @default.
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- W3100507412 date "2009-12-10" @default.
- W3100507412 modified "2023-09-30" @default.
- W3100507412 title "THE KAKEYA SET AND MAXIMAL CONJECTURES FOR ALGEBRAIC VARIETIES OVER FINITE FIELDS" @default.
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- W3100507412 doi "https://doi.org/10.1112/s0025579309000400" @default.
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