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F{\\o}lner showed that if $A \\subseteq G$ has positive upper Banach density, then $A - A$ contains an almost Bohr set -- a set of the form $B \\setminus E$ where $B$ is a Bohr set and $E$ has zero Banach density. We study the sets $S \\subseteq G$ for which $A - A + S$ contains a Bohr set for every $A \\subseteq G$ of positive upper Banach density. For $G = \\mathbb{Z}$, we show that the sets $\\{n^2: n \\in \\mathbb{N}\\}$, $\\{p - 1: p \\text{ prime}\\}$, and $\\{ \\lfloor n^c \\rfloor: n \\in \\mathbb{N} \\}$ with $c > 0$, have this property. 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