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Source. Conjecture 6 and the paragraph after it, p. 4, with Theorem 5, p. 4, of Javier Cilleruelo and Andrew Granville, Lattice points on circles, squares in arithmetic progressions and sumsets of squares, Additive Combinatorics, CRM Proceedings and Lecture Notes 43 (Amer. Math. Soc., 2007), 241-262. Labels and pages are those of the arXiv preprint math/0608109v1 identified on the source card.

Setting

An affine cube of dimension $d$ in $\mathbb Z$ is, in the paper's definition (p. 4), a set of integers

$$\Bigl\{\,b_0+\sum_{i\in I}b_i:\ I\subset\{1,\ldots,d\}\Bigr\}$$

for non-zero integers $b_0,\ldots,b_d$.

Statement

Conjecture 6 (Solymosi; p. 4, quoted). "There exists an integer $d>0$ such that there is no affine cube of dimension $d$ of distinct squares."

The paper attributes the conjecture to Solymosi's Elementary additive combinatorics in the same volume, and states it as a conjecture; it is not proved here.

Remark after the conjecture (p. 4). The paper notes that the conjecture follows from the Bombieri-Lang conjecture. Its argument: in an affine cube of dimension $d$ of distinct squares, every $x^2$ in the subcube $\{b_0+\sum_{i\in I}b_i: I\subset\{3,\ldots,d\}\}$ has $x^2+b_1$, $x^2+b_2$ and $x^2+b_1+b_2$ also square, so at least $2^{d-2}$ integers $x$ make $f(x)=(x^2+b_1)(x^2+b_2)(x^2+b_1+b_2)$ a square, and $2^{d-2}\le B$ with $B$ the uniform bound used in the proof of Theorem 2 (from Caporaso, Harris and Mazur, for polynomials of degree five or six without repeated roots). That $f$ has no repeated root follows because distinctness of the cube's elements makes $b_1$, $b_2$ and $b_1+b_2$ distinct and non-zero (an observation of this page; the paper does not spell it out).

Theorem 5 (p. 4). Conjecture 6 implies that there exists $\delta>0$ for which $|A+A|\gg|A|^{1+\delta}$. The paper introduces it as a weak version of Ruzsa's conjecture (its Conjecture 5, on finite sets of squares) and derives it from Solymosi's theorem that a set $A$ of reals with $|A+A|\ll_d|A|^{1+1/(2^{d-1}-1)}$ contains many affine cubes of dimension $d$; the flowchart on p. 10 records it as "Conjecture 5 for some $\varepsilon<1$".

Section 8 (p. 17) recalls the conjecture as the claim that there are no generalized arithmetic progressions of squares with every $J_i=2$ and $d$ sufficiently large.

Proof pointer

The paper gives no proof of the conjecture. The derivation from Bombieri-Lang is the paragraph after it on p. 4, resting on the proof of Theorem 2 (p. 3).

Dependencies

The Bombieri-Lang conjecture (unproved), through Caporaso, Harris and Mazur, Uniformity of rational points, J. Amer. Math. Soc. 10 (1997), 1-35, as used in the proof of Theorem 2.

Read depth: claims checked; the conjecture, the remark and Theorem 5 were read clause by clause on p. 4.

Bears on

• Problem 782: the problem's second question asks whether the squares contain arbitrarily large cubes $a+\{\sum_i\epsilon_ib_i:\epsilon_i\in\{0,1\}\}$. Conjecture 6 says they do not, for cubes with non-zero $b_0,\ldots,b_d$ whose $2^d$ elements are distinct squares. The paper does not prove the conjecture; it derives it from the unproved Bombieri-Lang conjecture. The paper does not discuss the problem's first question, on quasi-progressions.