The answer is no by Ruzsa [Ru87], who proved that if $A$ is an essential component then there exists some constant $c>0$ such that $\lvert A\cap \{1,\ldots,N\}\rvert \geq (\log N)^{1+c}$ for all large $N$.

Furthermore, Ruzsa proves that this is best possible, in that for any $c>0$ there exists an essential component $A$ for which $\lvert A\cap \{1,\ldots,N\}\rvert \leq (\log N)^{1+c}$ for all large $N$.

See also [1146] for whether $\{2^m3^n\}$ is an essential component.

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This page was last edited 23 January 2026.