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Peres and Solomyak asked the question: Do there exist self-similar sets withpositive Lebesgue measure and empty interior? This question was answered in theaffirmative by Cs\"{o}rnyei et al. They gave a parameterised family of iteratedfunction systems for which almost all of the corresponding self-similar setssatisfied the required properties. They do not however provide an explicitexample. Motivated by a desire to construct an explicit example, we in thispaper provide an explicit construction of an infinitely generated self-similarset with positive Lebesgue measure and empty interior.

An infinitely generated self-similar set with positive Lebesgue measure and empty interior