For \(\lambda\in(0,1)\), let \(D(\lambda)=\{d_{1}(\lambda),\dots,d_{m}(\lambda)\}\) be a finite set of digits depending on \(\lambda\). Denote \(\Omega:=\{1,\dots,m\}^{\mathbb Z_{+}}\), and equip \(\Omega\) with the Bernoulli product measure \(\mu=p^{\mathbb Z_+}\) for a given probability vector \(p=(p_{1},\dots,p_{m})\). Define the map \(\Pi_{\lambda}:\Omega\to\mathbb{R}\) by \[ \Pi_{\lambda}(\omega) := \sum_{j=0}^{\infty} d_{\omega_{j}}(\lambda) \lambda^j. \] This paper is concerned with the self-similar sets \(C_{\lambda}:=\Pi_{\lambda}(\Omega)\) and measures \(\nu_{\lambda}(D,p) := \mu\circ\Pi_{\lambda}^{-1}\). For \(m=2\) and \(\lambda\in(0,1/2)\), the set \(C_{\lambda}\) is a classical Cantor set. One would expect that the arithmetic sum of two such sets should have positive Lebesgue measure if the sum of their Hausdorff dimensions exceeds 1, but there are known counterexamples, e.g., the arithmetic sum of \(C_\lambda\) with itself is a Lebesgue null set for \(m=2\) and all \(\lambda\in(1/4,1/3)\). Here the authors show that this behaviour is exceptional: If the family \(\{C_\lambda\}_{\lambda\in J}\) satisfies a certain ``strong separation condition'' on an interval \(J\subseteq(0,1)\), then for any fixed compact \(K\subseteq\mathbb{R}\), the arithmetic sum \(K+C_\lambda\) has positive Lebesgue measure for a.e.\ \(\lambda \in J\) with \(\dim_{H} K + \dim_{H}C_{\lambda}>1\), and satisfies \(\dim_{H}(K+C_{\lambda}) = \dim_{H}K+\dim_{H}C_{\lambda}\) for a.e.\ \(\lambda\in J\) with \(\dim_{H} K + \dim_{H}C_{\lambda}\prod_{i=1}^{m} p_{i}^{p_{i}}\) and singular for all \(\lambda\in J\) with \(\lambda<\prod_{i=1}^{m} p_{i}^{p_{i}}\). In the case of absolute continuity they give conditions for the Radon-Nikodým derivative of \(\nu_\lambda\) to be in some \(L^q\); this latter approach is entirely new. Anyone interested in these topics is strongly encouraged to study this paper more closely; it is very rich in ideas, results and stimuli for further research.