Uniform half-circle balance for independent Bernoulli variables

Let $n$ be even and let $X_1,\dots ,X_n$ be independent Bernoulli variables with parameter $1/2$, arranged on a circle. The probability that some consecutive half-circle $I$ has $|{\sum }_{i\in I}{X}_i-{\sum }_{i\notin I}{X}_i|\ge w$ is at most $2n\exp (-2w^2/n)$. For a fixed $I$, the variable $Z_I={\sum }_{i\in I}{X}_i+{\sum }_{i\notin I}(1-X_i)$ has the binomial distribution $\mathrm{Bin}(n,1/2)$, and the signed imbalance is $Z_I-n/2$. Apply Hoeffding inequality to this single binomial variable, then take a union bound over the at most $n$ intervals. Consequently $w=c\sqrt{n\log n}$ suffices for uniform balance with probability tending to one whenever $c>1/\sqrt{2}$.