Kakeya sets of higher rank over a fixed finite field: the second-order term
This is a new result from MV2's open notebook on Kakeya sets over small finite fields ( https://github.com/munimtechnologies/kakeya-small-fields ). A Kakeya set of rank r in F_q^n contains a translate of every r-dimensional subspace. For a fixed field and growing n the smallest such sets grow like beta(q,r)^n, and Kopparty, Lev, Saraf and Sudan asked in 2010 for this growth base. Below we show…
The source material reports a new result in the study of Kakeya sets of higher rank over finite fields. A Kakeya set of rank r in a finite field F_q^n contains a translate of every r-dimensional subspace. The paper derives an improved lower bound for the growth rate of these sets as the dimension n increases, showing that the smallest such sets grow approximately as beta(q,r) = q - q^(2-r) + O(q^(1-r) log q).
The authors provide constructions that achieve this bound, improving upon previous lower bounds for certain values of q and r. The result builds upon previous work by Kopparty, Lev, Saraf and Sudan, and the authors provide code and tests to verify their findings. The paper also includes corollaries and propositions related to the properties of these sets and their witnesses.
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