Abstract
The arithmetic Kakeya conjecture, formulated by Katz and Tao (Math Res Lett 6(5–6):625–630, 1999), is a statement about addition of finite sets. It is known to imply a form of the Kakeya conjecture, namely that the upper Minkowski dimension of a Besicovitch set in {mathbf {R}}^n is n. In this note we discuss this conjecture, giving a number of equivalent forms of it. We show that a natural finite field variant of it does hold. We also give some lower bounds.
Highlights
One of the main aims of this paper is to give a number of equivalent forms of the conjecture
Lim lim log Fk(N ) = 1. k→∞ N →∞ log N. This conjecture was raised by the second author as [17, Conjecture 4.2], but no links to the Kakeya problem were mentioned there
It follows that the expected number of pairs (i, j) with i < j for which φθ = φθ (d j ) is at most
Summary
We turn to Conjecture 1.4 (n). To demonstrate its equivalence to the first three conjectures, it suffices to show that for each n we have Conjecture 1’ ⇒ Conjecture 1.4 (n) ⇒. A2 contains a progression of length k and common difference d for pn distinct values of d. N } is either wholly contained in some I j , or else is split into two progressions, one in I j and the other in I j+1, with one of these having length at least k/2 It follows that the set A2 ⊂ Z defined by. Whilst A3 contains a progression of length k/2 and common difference d for all d in some set D ⊂ {0, . A4 contains a progression of length k and common difference d for all d ∈ π (n)(D), that is to say for n,k N n,k pn values of d. Containing a progression of length k and common difference d, for all d ∈ Fnp\{0}. It is quite interesting that the innocent-looking statement (3.5) implies two famous unsolved problems in completely different mathematical areas
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