Consider the parity function $MOD_2(x) = x_1 \oplus \cdots \oplus x_n$ for $x \in \mathbb{F}_2^n$. I am concerned about the degree bounds for a real polynomial $f$ which approximates $MOD_2$ well in the L1-norm i.e. $\frac{1}{2^n}\sum_{x \in \mathbb{F}_2^n} |f(x)-MOD_2(x)|$ is as low as possible.
I am aware of the fact that any real polynomial $f$ which satisfies $|f(x)-MOD_2(x)|\leq 1/3$ for all $x \in \mathbb{F}_2^n$ needs degree $\Theta(n)$. I also know that a real polynomial which agrees with the parity function on $2/3$rd of the inputs requires degree $\Theta(\sqrt{n})$. But the polynomial used to attain this degree upper bound is one obtained by interpolating the parity function over all inputs $x$ whose hamming weight lies in $[n/2-\sqrt{n}, n/2+\sqrt{n}]$. The problem with this polynomial is that it can take huge values on inputs where it disagrees with the parity function and hence the L1-error $\frac{1}{2^n}\sum_{x \in \mathbb{F}_2^n} |f(x)-MOD_2(x)|$ can potentially be as high as $O(n^{\sqrt{n}})$. What are the best known upper and lower bounds on the degree of a polynomial approximating $MOD_2$ when we want the L1-error to be $\leq \frac{1}{3}$?
