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Setting

Section 2 fixes $k\ge4$ and writes $X\ll Y$, $X=O(Y)$ for $|X|\le C_kY$ with $C_k$ depending only on $k$, and $X\asymp Y$ for $X\ll Y\ll X$ (pp. 5--6). Boldface letters are random variables; $\mathbb P$ is probability.

Statement

Proposition 2.1 (Probabilistic construction, p. 6). Let $N$ be sufficiently large. Then there are a random tuple $(\mathbf n_1,\ldots,\mathbf n_k)$ of natural numbers and an event $E$ such that:

(i) with probability $1$, the product $\mathbf n_1\cdots\mathbf n_k$ is a perfect square;

(ii) on $E$, $\mathbf n_i\le N$ for every $i=1,\ldots,k$;

(iii) $\mathbb P(E)\gg1/\log^kN$;

(iv) for every $1\le i<j\le k$, $\mathbb P(\mathbf n_i=\mathbf n_j)=o(1/\log^kN)$ as $N\to\infty$;

(v) for every $1\le n\le N$ and every $i=1,\ldots,k$, $\mathbb P(\mathbf n_i=n\wedge E)\ll1/(N\log^kN)$.

Proof pointer

Pp. 6--11. Each $\mathbf n_i$ is a product over the pairs $\{i,j\}$, $j\neq i$, of independent copies $\mathbf d_{i,j}$ and $\mathbf p_{i,j}$, so that every factor appears in exactly two of the $\mathbf n_i$ and the product is a square, the model of the factorization (1.4) on p. 5. Here $\mathbf d$ is a squarefree number with all prime factors below $N^{\varepsilon^2}$, weighted by $1/((k-1)^{\omega(d)}d)$, and $\mathbf p$ is a prime between $N^\varepsilon$ and $N$, weighted by $1/p$ (p. 6). The estimates use Mertens' theorems and the prime number theorem, and the bounds (iii) and (v) are double counting arguments over polytopes of logarithmic sizes, integrated by the Fubini--Tonelli theorem (pp. 8--11). Each count rests on the linear independence of a family of linear forms: on p. 9 this needs only $k\ge3$, while the step for (v) on p. 10 applies the same argument with $k-1$ in place of $k$ and is where the hypothesis $k\ge4$ is used. Property (iv) is a union bound (p. 8). The paper's Theorem 1.2 follows from (i)--(v) on p. 6, as the Theorem 1.2 page explains.

Remarks in the paper

Remark 2.3 (p. 11, suggested by Andrew Granville) sketches a modification showing that a set $A\subseteq\{1,\ldots,N\}$ of at least $(1-c_k)N$ elements, for $k\ge4$, $N$ large and $c_k>0$ small, contains at least $N^{k/2}/\log^{k\log(k-1)+o(1)}N$ tuples of $k$ distinct elements multiplying to a square. Remark 2.4 (p. 12) sketches the extension to $m$-th powers for $m\ge2$ and $k\ge m+2$, through an analogue of this proposition. Both are sketches; neither was checked here.

Read depth

Claims checked: the statement was read clause by clause on the page image of the print; the proof on pp. 6--11 was read for structure only. Nothing here is independently reviewed.

Source. Terence Tao, On product representations of squares, Acta Math. Hungar. 175 (2025), no. 1, 142--157, doi:10.1007/s10474-025-01505-7; preprint arXiv:2405.11610. Labels and pages are those of arXiv:2405.11610v3, the edition named on the source card.

Bears on

• Problem 121: the proposition is the construction from which the paper derives Theorem 1.2, the negative answer to the problem's questions; on its own it states nothing about $F_k(N)$.