Extremal Problems for GCDs and LCMs in Higher Dimensions

TL;DR

This paper investigates extremal problems involving GCDs and LCMs of integer tuples in higher dimensions, providing bounds and methods that extend previous work and are nearly optimal.

Contribution

It extends extremal GCD and LCM bounds to higher dimensions, introducing new techniques and establishing near-optimal estimates.

Findings

Derived bounds for GCD-based problems in higher dimensions.

Established bounds for LCM-based problems in higher dimensions.

Proved the near-optimality of the obtained bounds.

Abstract

We study extremal problems for tuples of integers chosen from sets $A_i \subset [X_i,2X_i]$ for $1\le i\le k$, under large GCD and small LCM conditions. For the GCD problem, we extend the work of Green and Walker to higher dimensions. Specifically, for $k\ge 3$, if $\gcd(a_1,\dots,a_k)\ge D$ for at least a proportion $\delta$ of the tuples in $\prod_{i=1}^k A_i$, then $$ \prod_{i=1}^k |A_i| \ll_{k,\varepsilon} \delta^{-k/(k-1)-\varepsilon} \frac{\prod_{i=1}^k X_i}{D^k}. $$ The proof is based on a minimal counterexample argument and a new high-dimensional measure concentration lemma. We also establish a large sieve-type inequality to obtain a complementary estimate for the GCD problem. For the LCM problem, we use a quite different method to show that, for all $k\ge 2$, $$ \prod_{i=1}^k |A_i| \ll_{k,\varepsilon} \delta^{-k/(k-1)} \frac{L^{k/(k-1)+\varepsilon}} {\bigl(\prod_{i=1}^k X_i\bigr)^{1/(k-1)}}, $$ whenever $\operatorname{lcm}(a_1,\dots,a_k)\le L$ for at least a proportion $\delta$ of the $k$-tuples in $\prod_{i=1}^k A_i$. Finally, we show that these bounds are essentially best possible up to $\varepsilon$-losses in the exponent.