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Source. Theorem 1.3, p. 5, with its proof in Section 6, pp. 17--20, 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

$\mathbb F_p^\infty$ is the vector space over $\mathbb F_p$ of countably infinite dimension, and $\mathbf H$ is Shannon entropy.

Theorem 1.3 (p. 5). Let $X$ and $Y$ be $\mathbb F_p^\infty$-valued random variables, each taking only finitely many values. Then

$$\mathbf H(X-Y)\le\Bigl(1+O\bigl(\tfrac{1}{\log p}\bigr)\Bigr) \sup_{r\in\mathbb F_p\cup\{\infty\}\setminus\{-1\}}\mathbf H(X+rY),$$

where the constant in the $O(\cdot)$ is absolute.

This is the finite field variant of the paper's Conjecture 2 (see Theorem 1.1). The paper says the $O(1/\log p)$ term is best possible (p. 5); the example on pp. 19--20, with $X=(a+b,ab)$ and $Y=(a+b',ab')$ for independent uniform $a,b,b'\in\mathbb F_p$, has $\mathbf H(X-Y)=2\log p+O(\log p/p)$ and $\mathbf H(X+rY)\le2\log p-\log2+O(\log p/p)$ for $r\ne-1$. The remark introducing that example names Theorem 1.2 [sic]; it concerns Theorem 1.3.

Proof pointer

Section 6, pp. 17--20. Assuming $\mathbf H(X-Y)\ge(1+\varepsilon)\sup\mathbf H(X+rY)$, a tensor-power construction gives large finite $B\subset\mathbb F_p^\infty\times\mathbb F_p^\infty$ with $\#\pi_{-1}(B)\ge\sup_{r\ne-1}(\#\pi_r(B))^{1+\varepsilon/2}$; the lines through the pairs of $B$ form a set $A$ of size at most $p\sup\#\pi_r(B)$ containing a line with common difference $d$ for every nonzero $d\in\pi_{-1}(B)$, so $N=\#\pi_{-1}(B)-1$ directions. Proposition 6.1 (p. 18), proved by a random projection to $\mathbb F_p^n$, covering by translates (Corollary A.3, p. 20) and a lower bound for finite field Kakeya sets (the paper's reference [6]), gives $\#A\gg_pN^{1-\log2/\log p-o(1)}$ for such sets, forcing $\varepsilon=O(1/\log p)$.

Read depth

Claims checked: the statement, the remark on sharpness and the outline of Section 6 were read on the print; the finite field Kakeya bound cited as reference [6] was not read. Nothing here is independently reviewed.

Dependencies

External input: the finite field Kakeya bound of the paper's reference [6].

Bears on

No Erdős problem in the corpus.