Let \(g(x, y) \in \mathbb {Z}[x, y]\) . We say that (x, y) is a rational solution to \(g(x, y) = 0\) if \((x, y) \in \mathbb {Q} \times \mathbb {Q}\) . A pair \( (x, y) \in \mathbb {R}^2 \) is called a non-trivial solution of \( g \) if \( g(x, y) = 0 \) and at least one of \( x \) or \( y \) is irrational, i.e., \( x \notin \mathbb {Q} \) or \( y \notin \mathbb {Q} \) . In this paper, we investigate rational and non-trivial solutions to certain classical algebraic equations within Cantor sets. Under specific conditions, we prove that many classical algebraic equations have only trivial rational solutions in fractal sets. By combining a variant of the thickness theorem, we establish the existence of non-trivial solutions to these equations. For example, we demonstrate that there exist at least 10,000,000 distinct pairs \((x_i, y_i)\) from the Cartesian product of the middle-third Cantor set such that \(\begin{aligned} x_i^2 + y_i^2 = 1, \end{aligned}\) where \(x_i \notin \mathbb {Q}\) . Our main approach relies on techniques from q-expansions.
