$\def\N{\mathbb{N}}$For any set $S\subseteq\N$ and $a\in \N$, we let $a+S =\{a+s: s\in S\}$. Are there infinite sets $A, S\subseteq \N$ such that $\{a+S: a\in A\}$ is an infinite partition of $\N$?
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2$\begingroup$ To make things more symmetrical, does there exists $A,B\subset\mathbb N$ such that every $n\in\mathbb N$ can be written uniquely as $n=a+b$ with $a\in A$ and $b\in B$? $\endgroup$Corentin B– Corentin B2025-05-23 09:22:44 +00:00Commented May 23 at 9:22
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$\begingroup$ Very pretty @CorentinB! $\endgroup$Dominic van der Zypen– Dominic van der Zypen2025-05-23 11:30:49 +00:00Commented May 23 at 11:30
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Yes, $A=$numbers with non-zero binary digits on even places, $S=$numbers with non-zero binary digits on odd places
