Extremal Problems for GCDs and LCMs in Higher Dimensions

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 $δ$ of the tuples in $\prod_{i=1}^k A_i$, then $$ \prod_{i=1}^k |A_i| \ll_{k,\varepsilon} δ^{-k/(k-1)-\varepsilon} \frac{\prod_{i=1}^k X_i}{D^k}. $$ 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 $k\ge 2$, $$ \prod_{i=1}^k |A_i| \ll_{k,\varepsilon} δ^{-k/(k-1)} \frac{L^{k/(k-1)+\varepsilon}} {\bigl(\prod_{i=1}^k X_i\bigr)^{1/(k-1)}}, $$ whenever $\operatorname{lcm}(a_1,\dots,a_k)\le L$ for at least a proportion $δ$ 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.