Question 71
We say that a doubly infinite sequence
$$
\dots, s_{-2}, s_{-1}, s_0, s_1, s_2, \dots
$$
is subaveraging if $s_n = (s_{n-1} + s_{n+1})/4$ for all integers $n$.
a. Find a subaveraging sequence in which all entries are different from each other. Prove that all entries are indeed distinct.
b. Show that if $(s_n)$ is a subaveraging sequence such that there exist distinct integers $m, n$ such that $s_m = s_n$, then there are infinitely many pairs of distinct integers $i, j$ with $s_i = s_j$.
$$
\dots, s_{-2}, s_{-1}, s_0, s_1, s_2, \dots
$$
is subaveraging if $s_n = (s_{n-1} + s_{n+1})/4$ for all integers $n$.
a. Find a subaveraging sequence in which all entries are different from each other. Prove that all entries are indeed distinct.
b. Show that if $(s_n)$ is a subaveraging sequence such that there exist distinct integers $m, n$ such that $s_m = s_n$, then there are infinitely many pairs of distinct integers $i, j$ with $s_i = s_j$.