For given $n$, consider a polynomial $\sum\_{k=0}^na\_kz^k$ with all coefficients $a\_k\in\{\pm1\}$. I am interested in the following: 



> 
> **How big can the modulus of a non-real root of such a polynomial be?** 
> 
> 
> 


Wlog we can assume $a\_n=1,a\_{n-1}=-1$. Then a systematic search for small $n$, looking each time at the 10 or so extremal polynomials, exhibits a clear pattern for the highest coefficients: they come in groups of $3$ or $4$ consecutive +1's or -1's, starting with $z^n-z^{n-1}+++----+++---++++\cdots$. More precisely, the group sizes are displayed here (first line +1's, second line -1's)



```
1 3 3 4 3 3 3 4 3 3 3 4 3 3 4 4 3 3 4 4 3 3 4 4 3 3 4 . . .
 1 4 3 3 4 4 3 3 4 4 3 3 4 3 3 3 4 3 3 3 4 3 3 3 4 3 . . .

```

Note that there are never two adjacent groups of size 4 or three of size 3, so the pattern is rather regular, as may be expected. 


As $n$ grows, the extremal root $z\_0$ (and $\bar z\_0$) of the extremal polynomials does not at all jump around, but converges quite rapidly towards $0.93757749648487973269811306454355 \pm 1.2634174429011374851417570421775\; i$, e.g. we have 


$n=20\implies z\_0\approx 0.937537 \pm 1.26337\;i$ 


$n=40\implies z\_0\approx 0.9375774916 \pm 1.263417437\;i$ 


$n=60\implies z\_0\approx 0.9375774964839 \pm 1.26341744290078\;i$.


$n=80\implies z\_0\approx 0.93757749648487963 \pm 1.263417442901137422\;i$. 


$n=100\implies z\_0\approx 0.937577496484879732688 \pm 1.263417442901137485132\;i$. 


The obvious questions:



> 
> * **Can the above 3-4-sequences be characterized? E.g. are they periodic? or self-similar?**
> * **What can be said about the limit value of $z\_0$? Is there a (closed form or whatever) formula for $z\_0$?**
> 
> 
> 


Although the problem is not directly related to *Garsia numbers* (see the article mentioned [here](https://mathoverflow.net/questions/85324/bound-for-zeros-of-a-polynomial-with-bounded-integer-coefficients/85326#85326)), it naturally leads again to the question of limit points of zeros of such polynomials, given that $z\_0$ visibly yields one of those:



> 
> * **Are there only countably many non-real limit points of zeros of $\pm1$-polynomials?**
> 
> 
> 



$\pm1$-polynomials with a maximal non-real root
A closely related problem was considered by [Odlyzko and Poonen](http://retro.seals.ch/digbib/view;jsessionid=508577B1E580B9C6A779A3ECA50D6EF6?rid=ensmat-001:1993:39::181) who looked at the class of polynomials with $0$ or $1$ coefficients. On page 330, they discuss the question of the smallest (in size) non-real root in this family (which is of course equivalent to the largest non-real root) and observe that numerically its size is about $0.734$. In Section 6 of their paper they discuss computations plus a simple Rouche's theorem argument which would enable one to compute this as accurately as desired. The same arguments should carry over to this situation. There doesn't seem to be any more natural description of the smallest size of non-real roots in the Odlyzko-Poonen paper. 


