Recently The Vee confessed:
Literally every time I'm serving some soup I'm thinking of this little mathematical problem I devised.
Solving the problem, I introduced the following notion.
A number $q\in (1/2, 1)$ is approximating, if there exist non-negative numbers $A$ and $N$ such that for each $x_0\in [0,A]$ there exist $n\le N$ and a polynomial $P(x)$ of the form $\sum_{i=1}^n \pm x^i$ such that $P(1)=0$ and $|x_0-P(q)|\le Aq^n$. We can show that The Vee’s problem has a solution for each approximating number $q$. This poses my question
Which $q\in (1/2,1)$ are approximating?
Thanks.
My try.
I wrote a short paper on The Vee's problem and approximating numbers. There is shown that each $q\in (q_\infty,1)$ is approximating, where $q_\infty=0.5845751\dots$ is a unique positive root of a polynomial $Q_\infty(x)=x^4+x^3+2x^2-1$.
I am going to give a talk on the paper at Western Ukrainian online mathematical seminar at Monday, February 1, at 15:05 GMT+2. I hope the talk will be recorded and then put at Youtube.