An elementary question of Erdős and Graham

TL;DR

Let $A_k = \{ r(k-r) : 1 \le r \le k-1 \}$ and address Erdős–Graham's question on the size of $|A_n \cap A_m|$ with an elementary injection into the set $T_{m,n}$ of factor-pair representations of $m^2-n^2$. The bound $|A_n \cap A_m| \le \tau_{m,n}$, where $\tau_{m,n} = |T_{m,n}|$, is sharpened by a parity condition (equality when $m$ is even and $n$ is odd), and a constructive argument yields infinite sequences with any prescribed intersection size $s$. The paper also connects these arithmetic bounds to sum-product questions, deriving a conditional lower bound $\max\{|A+A|,|AA|\} \gg n^{4/3 - 3/(\log\log n)^{1-c}}$ for certain growth-restricted sets $A$, using an elementary, arithmetic-structure approach rather than geometric higher-energy methods. Overall, the work provides a tightly focused, fully arithmetic treatment of Erdős–Graham-type intersections and their implications for sum-product phenomena, offering sharp integer-structure insights complementing existing geometric results.

Abstract

Let $A_k=\{r(k-r): 1\leq r \leq k-1\}$. Erd\H os and Graham asked about the cardinality of the set of common elements. We answer this elementary question and apply our result to a sum-product type result.

Theorems & Definitions (4)

• proof : Proof of Theorem 1.1
• proof : Proof of Corollary 1.2
• proof : Proof of Theorem 1.3
• proof