Problem 226

Is there an entire non-linear function $f$ such that, for all $x\in\mathbb{R}$ , $x$ is rational if and only if $f(x)$ is?

Source: erdosproblems.com/226

Analysis

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

PROVED (LEAN). The site credits Barth and Schneider [BaSc70], whose theorem for arbitrary countable dense subsets of the reals gives the function at $A=B=\mathbb Q$ ; the claim page Barth and Schneider 1970 records it, accepted on its refereed publication and the site's credit. The site's Lean qualification refers to Boris Alexeev's formalization of that solution, linked from the claim page and not built by this corpus.

BaSc70: Barth, K. F. and Schneider, W. J., Entire functions mapping countable dense subsets of the reals onto each other monotonically. J. London Math. Soc. (2) (1970), 620-626.
BaSc71: Barth, K. F. and Schneider, W. J., Entire functions mapping arbitrary countable dense sets and their complements onto each other. J. London Math. Soc. (2) (1971/72), 482-488.
Ha74: Hayman, W. K., Research problems in function theory: new problems. (1974), 155-180.

Problem 226

Status

Proof claims

Discussion