For sets A_i subset [X_i,2X_i], if gcd of k-tuples is at least D for proportion delta then product |A_i| is at most delta to the power -k/(k-1)-eps times product X_i over D^k (k>=3); analogous bound holds for small lcm, and both are essentially optimal up to eps.
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Extremal Problems for GCDs and LCMs in Higher Dimensions
For sets A_i subset [X_i,2X_i], if gcd of k-tuples is at least D for proportion delta then product |A_i| is at most delta to the power -k/(k-1)-eps times product X_i over D^k (k>=3); analogous bound holds for small lcm, and both are essentially optimal up to eps.