Green 2017 arithmetic kakeya conjecture katz tao

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proposition_4_1: States that G_k(N), the least number over all primes p_1 < ... < p_N and intervals of length k p_N of integers in the interval divisible by some p_i, satisfies F'_k(N) <= G_k(N) <= k F'_k(N), so Conjectures 1' and 5 are equivalent.

theorem_1_1: States that Conjectures 1, 2, 3, 4(n) for each positive integer n, and 5 of Green and Ruzsa, five formulations of the Katz-Tao arithmetic Kakeya conjecture, are all equivalent.

theorem_1_2: States that the least size F_k(N) of a set of integers containing a k-term progression with each common difference 1, ..., N satisfies lim_N log F_k(N)/log N <= 1 - c/log log k for an absolute constant c > 0.

theorem_1_3: States that for finitely-valued random variables X, Y in the vector space over F_p of countably infinite dimension, H(X - Y) is at most 1 + O(1/log p) times the supremum of H(X + rY) over r in F_p and infinity other than -1, with an absolute implied constant.

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Ben Green, Imre Ruzsa, On the arithmetic Kakeya conjecture of Katz and Tao. arXiv preprint (2017). arXiv:1712.02108. The copy read for this card is arXiv:1712.02108v1 (6 December 2017), 22 pages, the only version. The arXiv record names arXiv's non-exclusive distribution license (https://arxiv.org/abs/1712.02108), recorded as reserved.

Green and Ruzsa discuss the arithmetic Kakeya conjecture of Katz and Tao, a purely additive statement about sums of finite sets which implies that Besicovitch sets in R^n have full upper Minkowski dimension. Theorem 1.1 shows that five formulations are equivalent: Conjecture 1 (writing F_k(N) for the least size of a set of integers containing a k-term progression with common difference d for each d = 1, ..., N, lim_k lim_N log F_k(N)/log N = 1), an entropy form, the original Katz-Tao form, a family of finite field forms, and Conjecture 5, a lower bound >>_k N^{1 - gamma_k} with gamma_k -> 0 as k -> infinity for how many integers in an interval of length k p_N are multiples of one of the primes p_1 < ... < p_N. The paper says Conjecture 5 relates closely to a question of Erdos and Selfridge, who asked whether one can take gamma_k = 0. Theorem 1.2 shows that lim_N log F_k(N)/log N <= 1 - c/log log k for an absolute constant c > 0, so any convergence in Conjecture 1 is slow; Theorem 1.3 proves the entropy form for F_p^infinity-valued random variables with a factor 1 + O(1/log p). Proposition 4.1 gives F'_k(N) <= G_k(N) <= k F'_k(N), where G_k(N) is the least number, over all primes p_1 < ... < p_N and intervals of length k p_N, of integers in the interval divisible by some p_i, and F'_k(N) is the least size of a set containing k-term progressions with N different common differences; in particular Conjectures 1' and 5 are equivalent. The remark after Conjecture 5 notes that Proposition 4.1 and Theorem 1.2 together force gamma_k >> 1/log log k in Conjecture 5. In the discussion thread of problem 1143 Thomas Bloom pointed to Proposition 4.1 as the link between that problem's long intervals (alpha > 3) and the arithmetic Kakeya conjecture.

Source: https://arxiv.org/abs/1712.02108.

Result pages, labels and pages from the arXiv print:

• Theorem 1.1 (p. 4): Conjectures 1, 2, 3, 4($n$) for each $n$, and 5 are all equivalent; the five conjectures are stated on pp. 2--3.
• Theorem 1.2 (p. 4): $\lim_N\log F_k(N)/\log N\le1-c/\log\log k$ with $c>0$ absolute.
• Theorem 1.3 (p. 5): the entropy inequality of Conjecture 2 over $\mathbb F_p^\infty$ with factor $1+O(1/\log p)$.
• Proposition 4.1 (p. 14): $F'_k(N)\le G_k(N)\le kF'_k(N)$.

Read status. Claims checked for the four results above, read clause by clause on the print; the proofs of Theorem 1.2 and Proposition 4.1 were followed, and the external theorems they cite were not read.

Bears on. #1143: for a fixed integer alpha = k, G_k(N) is the least value of that problem's count over N primes and intervals of length k p_N. Proposition 4.1 with Theorem 1.1 makes the lower bound G_k(N) >>_k N^{1 - gamma_k} with gamma_k -> 0 as k -> infinity (Conjecture 5) equivalent to the arithmetic Kakeya conjecture, which the paper leaves open, and with Theorem 1.2 Proposition 4.1 gives G_k(N) <= k N^{1 - c/log log k + o(1)} as N -> infinity, so the count for fixed k can fall well below N. It settles no exact value of the count.

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