This question was previously posted on MSE https://math.stackexchange.com/questions/3305781/estimating-the-measure-of-a-pre-image-of-a-polynomial

Let $\sigma := 2/(3\sqrt{3})$, be a real number. And consider the polynomial functions $P_i:[-1,1]^{i+1}\to\mathbb{R} $ defined in the following way:

- $P_0 (x_0) = x_0$;
- $P_1(x_0,x_1) = x_0^3 + \sigma\cdot x_1$;
- $\vdots$
- $P_{i+1}(x_0,\ldots,x_{i+1}) = (P_{i}(x_0,...,x_{i}))^3 + \sigma\cdot x_{i+1}$.

Note that there is only one real root $x^*:= x^*(\sigma)$ of the following cubic equation $$x^3 + \sigma = x. $$

Now, let $\varepsilon>0$ (small) and define $$Q_i(\varepsilon) := m(P_i^{-1}(x^* - \varepsilon,x^* +\varepsilon)), $$ where $m$ is the lebesgue measure (remember $P_i^{-1}(x_* - \varepsilon,x_* +\varepsilon) \subset [-1,1]$).

My Question:Is it possible estimate $Q_i(\varepsilon)$? Moreover, there exists $i_0>0$, such that $Q_i(\varepsilon)\geq Q_{i+1}(\varepsilon)$, $\forall$ $i>i_0$?