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Source. Theorem 1.2, p. 4, with its proof in Section 5, pp. 16--17, of Ben Green and Imre Z. Ruzsa, On the arithmetic Kakeya conjecture of Katz and Tao, arXiv:1712.02108 (2017); the edition read is named on the source card.

Statement

Notation (Conjecture 1, p. 2). $F_k(N)$ is the size of the smallest set of integers containing, for each $d\in\{1,\ldots,N\}$, a $k$-term arithmetic progression with common difference $d$.

Theorem 1.2 (p. 4). There is an absolute constant $c>0$ such that

$$\lim_{N\to\infty}\frac{\log F_k(N)}{\log N}\le1-\frac{c}{\log\log k}.$$

The theorem is printed with $\lim$; the proof (pp. 16--17) shows $F_k(N)\ll_kN^{1-c/\log\log k}$ for all $N$ once $k$ is sufficiently large, which gives the bound for the upper limit. The paper presents it as showing that the convergence in Conjecture 1, if it occurs, is very slow.

Proof pointer

Section 5, pp. 16--17. Let $Q$ be the product of the first $m=\lceil10\log k\rceil$ odd primes and let $S$ be the union of the progressions $\{x_d+jd:0\le j<k\}$, $1\le d<Q$, with $x_d\equiv d^2\pmod Q$. Completing the square shows that $x_d+jd$ takes at most $\frac12(p_i+1)$ values modulo each $p_i$, which gives $\#S\ll k^{-7}Q$ and so $\#S\le Q^{1-c/\log\log k}$ for large $k$. Base-$Q$ digit sets $\{s_0+s_1Q+\cdots+s_{n-1}Q^{n-1}:s_i\in S\}$ then handle every difference below $Q^n$, and taking $n$ minimal with $Q^n>N$ gives the bound.

Read depth

Claims checked: the statement and the proof on pp. 16--17 were read clause by clause on the print. Nothing here is independently reviewed.

Dependencies

None in the corpus.

Bears on

• Problem 1143: through Proposition 4.1 and $F'_k(N)\le F_k(N)$, the least count of that problem over $N$ primes for an integer $\alpha=k$ is at most $kN^{1-c/\log\log k+o(1)}$ as $N\to\infty$, and the paper records $\gamma_k\gg1/\log\log k$ in its Conjecture 5. An upper bound on the extremal count; it gives no lower bound.