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 $n\to \mathrm{\infty }$.

Let $\alpha$ be the golden ratio, $m\in \mathbb{N}$, and $B({\alpha }^m)$ the Beatty sequence (or Beatty set) generated by ${\alpha }^m$. In this article, we give some combinatorial structures of $B({\alpha }^m)$ and use them in the study of associated sumsets. In particular, we obtain, for each $m\in \mathbb{N}$, a positive integer $h=h(m)$ such that the $h$-fold sumset $hB({\alpha }^m)$ is a cofinite subset of $\mathbb{N}$. In addition, we explicitly give the integer $N=N(m)$ such that $hB({\alpha }^m)$ contains every integer that is larger than or equal to $N$, and show that this choice of $N$ is best possible when $m$ is small. We also propose some possible research problems. This paper extends the previous results on sumsets associated with upper and lower Wythoff sequences.