Powerfree integers and Fourier bounds

We develop a general approach for showing when a set of integers $\mathscr{A}$ has infinitely many $k^{th}$ powerfree numbers without relying on equidistribution estimates for $\mathscr{A}$. In particular, we show that if the Fourier transform of $\mathscr{A}$ satisfies certain $L^{\infty}$ and $L^{1}$ bounds, and is also "decreasing" in some sense, then $\mathscr{A}$ contains infinitely many $k^{th}$ powerfree numbers. We then use this method to show that there are infinitely many cubefree palindromes in base $b\ge 1100$, and in the process we obtain new $L^{1}$ bounds for the Fourier transform of the set of palindromes. We also show that there are infinitely many squarefree integers such that its reverse is also squarefree in any base $b\ge 2$. Moreover, we show that there are infinitely many squarefree integers with a missing digit in base $b\ge 5$, and infinitely many such cubefree integers in base $b\ge 3$.