Extremal orders and races between palindromes in different bases

Let $b\ge 2$ and $n\ge 1$ be integers. Then $n$ is said to be a palindrome in base $b$ (or $b$-adic palindrome) if the representation of $n$ in base $b$ reads the same backward as forward. Let $A_b(n)$ be the number of $b$-adic palindromes less than or equal to $n$. In this article, we obtain extremal orders of $A_b(n)$. We also study the comparison between the number of palindromes in different bases and prove that if $b\ne b_1$, then $A_b(n)-A_{b_1}(n)$ changes signs infinitely often as $n\to \mathrm{\infty }$.