Extremal orders and races between palindromes in different bases

Let $ b \geq 2 $ and $ n \geq 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 \neq b_{1} $, then $ A_{b} (n) - A_{b_{1}} (n) $ changes signs infinitely often as $ n \rightarrow \infty $.