We prove that for any integers $\alpha, \beta > 1$, the existential fragment of the first-order theory of the structure $\langle \mathbb{Z}; 0,1,<, +, \alpha^{\mathbb{N}}, \beta^{\mathbb{N}}\rangle$ is decidable (where $\alpha^{\mathbb{N}}$ is the set of positive integer powers of $\alpha$, and likewise for $\beta^{\mathbb{N}}$). On the other hand, we show by way of hardness that decidability of the existential fragment of the theory of $\langle \mathbb{N}; 0,1, <, +, x\mapsto \alpha^x, x \mapsto \beta^x\rangle$ for any multiplicatively independent $\alpha,\beta > 1$ would lead to mathematical breakthroughs regarding base-$\alpha$ and base-$\beta$ expansions of certain transcendental numbers.

翻译：我们证明，对于任意整数 $\alpha, \beta > 1$，结构 $\langle \mathbb{Z}; 0,1,<, +, \alpha^{\mathbb{N}}, \beta^{\mathbb{N}}\rangle$ 的一阶理论存在片段是可判定的（其中 $\alpha^{\mathbb{N}}$ 表示 $\alpha$ 的正整数幂构成的集合，$\beta^{\mathbb{N}}$ 同理）。另一方面，我们通过难度分析表明：对于任意乘法独立的 $\alpha,\beta > 1$，若结构 $\langle \mathbb{N}; 0,1, <, +, x\mapsto \alpha^x, x \mapsto \beta^x\rangle$ 的理论存在片段具有可判定性，则将推动关于某些超越数的 $\alpha$ 进制与 $\beta$ 进制展开的数学突破。