---

Source. Theorem 1.1, p. 4, with Conjectures 1 to 5 on pp. 2--3 and Conjecture 1' on p. 5, 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

The five conjectures, in the paper's notation.

• Conjecture 1 (p. 2). For positive integers $k,N$ let $F_k(N)$ be the size of the smallest set of integers that contains, for each $d\in\{1,\ldots,N\}$, a $k$-term arithmetic progression with common difference $d$. Then

$$\lim_{k\to\infty}\lim_{N\to\infty}\frac{\log F_k(N)}{\log N}=1.$$

• Conjecture 2 (p. 2). For real-valued random variables $X,Y$ taking only finitely many values, and any $\varepsilon>0$, there are $r_1,\ldots,r_k\in\mathbb Q$, none equal to $-1$, with $\mathbf H(X-Y)\le(1+\varepsilon)\sup_j\mathbf H(X+r_jY)$, where $\mathbf H$ is the Shannon entropy. A footnote allows the $r_j$ to lie in $\mathbb Q\cup\{\infty\}$, with $X+\infty Y=Y$, and says the two versions are equivalent.
• Conjecture 3 (p. 3), the Katz--Tao form. For $A\subset\mathbb Z\times\mathbb Z$ finite and $r$ rational write $\pi_r(A)=\{x+ry:(x,y)\in A\}$ and $\pi_\infty(A)=\{y:(x,y)\in A\}$. For every $\varepsilon>0$ there are $r_1,\ldots,r_k\in\mathbb Q\cup\{\infty\}$, none equal to $-1$, such that $\#\pi_{-1}(A)\le\sup_i\#\pi_{r_i}(A)^{1+\varepsilon}$ for all finite $A\subset\mathbb Z\times\mathbb Z$.
• Conjecture 4($n$) (p. 3). For a positive integer $k$ and a prime $p$ let $f_{k,n}(p)$ be the size of the smallest set in $\mathbb F_p^n$ containing, for every $d\in\mathbb F_p^n\setminus\{0\}$, a $k$-term progression with common difference $d$. Then $\lim_{k\to\infty}\lim_{p\to\infty}\log f_{k,n}(p)/\log p=n$. (The displayed limit prints the subscript as $f_{n,k}(p)$.)
• Conjecture 5 (p. 3). Fix a positive integer $k$. Uniformly for all positive integers $N$, all sets of primes $p_1<\cdots<p_N$ and all intervals $I\subset\mathbb N$ of length $kp_N$,

$$\#\Bigl(I\cap\bigcup_{i=1}^N p_i\mathbb Z\Bigr)\gg_k N^{1-\gamma_k},$$

where $\gamma_k\to0$ as $k\to\infty$.

Theorem 1.1 (p. 4, quoted). "Conjectures 1, 2, 3, 4($n$) (for each $n = 1, 2, 3, \ldots$) and 5 are all equivalent."

The paper also uses (p. 5) Conjecture 1': with $F'_k(N)$ the size of the smallest $A\subset\mathbb Z$ containing a $k$-term arithmetic progression with common difference $d$ for $N$ different values of $d$, $\lim_{k\to\infty}\lim_{N\to\infty}\log F'_k(N)/\log N=1$.

Remarks the paper attaches (pp. 3--4). Conjecture 3, hence each of the others, is known to imply that every Besicovitch set in $\mathbb R^n$ has upper Minkowski dimension $n$. The equivalence of Conjectures 2 and 3 is attributed to the second author's earlier work. Erdős and Selfridge asked whether $\gamma_k=0$ is possible in Conjecture 5; the paper records that the answer is no, with $\gamma_k\ge1/k$ (crediting the second author's reference [16]), and that Proposition 4.1 and Theorem 1.2 together give $\gamma_k\gg1/\log\log k$.

Proof pointer

Section 2 (pp. 5--11) proves Conjectures 1, 1', 2 and 3 equivalent; Proposition 2.1 (p. 6) gives $F_k(N)\ll k^3\log N\cdot F'_k(N)$, so Conjectures 1 and 1' are equivalent since $F'_k(N)\le F_k(N)$. Section 3 (pp. 11--14) brings in the finite field forms Conjecture 4($n$). Section 4 (pp. 14--16) proves Proposition 4.1, which makes Conjectures 1' and 5 equivalent.

Read depth

Claims checked: the five conjectures, Conjecture 1', Theorem 1.1 and the remarks around them were read clause by clause on the print. The proofs of Sections 2 and 3 were not followed; the proof of Proposition 4.1 was. Nothing here is independently reviewed.

Dependencies

Proposition 4.1 for the equivalence with Conjecture 5.

Bears on

• Problem 1143: Conjecture 5 is a lower bound, for each integer $\alpha=k$, on that problem's count $F_{kp_N}(p_1,\ldots,p_N)$ minimised over $N$ primes; Theorem 1.1 makes the bound $\gg_k N^{1-\gamma_k}$ with $\gamma_k\to0$ equivalent to the arithmetic Kakeya conjecture, which the paper leaves open. It proves no bound on the count.