Consider the following sequence: $$f_{n+1}=(\lambda f_n + \frac{1}{2})\mod 1 - \frac{1}{2}.$$ For the initial $f_0$, you can select any non-zero number from the open interval $(-\tfrac{1}{2}, \tfrac{1}{2})$. The parameter $\lambda>1$ is set to be irrational.
Let's take the partial sums: $$s_n=\sum\limits_{i=0}^{n} f_i.$$ We will examine how the behavior of $s_n$ depends on $\lambda$. At first glance, it may appear similar to a random walk. Therefore, it might seem that the sequence of $s_n$ is unbounded with probability $1$.
However, when $\lambda<2$, this does not seem to be true. I have proven that, starting from a certain number $N$, all $f_n$ belong to the union of intervals $[-\tfrac{1}{2}, -\tfrac{1}{2}(2-\lambda)] \cup [\tfrac{1}{2}(2-\lambda), \tfrac{1}{2}]$.
A computer experiment conducted in Mathematica illustrates the following behavior for $\lambda =\sqrt{2}$ seems to be bounded
On the other hand, for $\lambda=\sqrt{2}+1>2$ the behavior changed looks like random walks. In these plots, blue points represent $s_n$, while orange points represent $f_n$.
Mathematica code:
lam = Sqrt[2];
(*interesing in irrational, bigger or less then 2 *)
f0 = 0.4; (*should be from -1/2 to 1/2*)
n = 8000;
setF = {f0}; (*array of f_n*)
tsum = f0; (*current s_n*)
psum = {f0}; (*array of partial sums*)
For [i = 1, i < n + 1, i++, fn = Mod[lam setF[[i]] + 1/2, 1] - 1/2;
tsum = tsum + fn; AppendTo[setF, fn]; AppendTo[psum, tsum]]
ListPlot[{psum, setF}]
Is there any techniques which allow to prove that?