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Source. Proposition 4.1, p. 14, with its proof on pp. 14--16, 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. $F'_k(N)$ (p. 5) is the size of the smallest set $A\subset\mathbb Z$ containing an arithmetic progression of length $k$ and common difference $d$ for $N$ different values of $d$.

Proposition 4.1 (p. 14). Let $G_k(N)$ be the minimum, over all intervals $I$ of length $kp_N$ and all choices of primes $p_1<\cdots<p_N$, of $\#\bigl(I\cap\bigcup_{i=1}^N p_i\mathbb Z\bigr)$. Then

$$F'_k(N)\le G_k(N)\le kF'_k(N).$$

In particular Conjectures 1' and 5 (see Theorem 1.1) are equivalent.

Combined with Theorem 1.2 and $F'_k(N)\le F_k(N)$ (p. 6), the upper bound gives $G_k(N)\le kN^{1-c/\log\log k+o(1)}$ as $N\to\infty$, with $c>0$ absolute; the paper states the consequence as $\gamma_k\gg1/\log\log k$ in Conjecture 5 (p. 4).

Proof pointer

Lower bound: the multiples of the $p_i$ in an extremal interval contain a $k$-term progression with difference $p_i$ for each $i$. Upper bound: take an extremal set $A$ of positive integers with progressions of differences $d_1,\ldots,d_N$; the theorem of Green and Tao cited as the paper's reference [10, Theorem 1.2] gives $u,v$ with all of $v,u+v,\ldots,d_Nu+v$ prime in a short range $[(1-\delta)X,X]$; set $p_i=d_iu+v$, choose $w$ by the Chinese remainder theorem with $p_i\mid w+ua_i$, and check that an interval of length $kp_N$ placed at $w$ meets each $p_i\mathbb Z$ exactly in the progression $w+ua_i+jp_i$, $0\le j<k$, inside $w+u\cdot A+\{0,v,\ldots,(k-1)v\}$, a set of at most $kF'_k(N)$ elements. The remark after the proof (p. 16) says simpler arguments would do at the cost of logarithmic losses.

Read depth

Claims checked: the definition of $G_k(N)$, the statement and the proof on pp. 14--16 were read clause by clause on the print. The cited theorem of Green and Tao was not read. Nothing here is independently reviewed.

Dependencies

External input: Green and Tao, the paper's reference [10], Theorem 1.2.

Bears on

• Problem 1143: for an integer $\alpha=k$, $G_k(N)$ is the least value of that problem's count $F_{kp_N}(p_1,\ldots,p_N)$ over $N$ primes. The proposition places it between $F'_k(N)$ and $kF'_k(N)$, so the problem's least count for fixed integer $\alpha$ is, up to a factor $k$, the arithmetic Kakeya quantity $F'_k(N)$; with Theorem 1.2 it is at most $kN^{1-c/\log\log k+o(1)}$. It settles no exact value or order of the count.