Critical exponents of infinite balanced words

Over an alphabet of size 3 we construct an infinite balanced word with critical exponent 2+sqrt(2)/2. Over an alphabet of size 4 we construct an infinite balanced word with critical exponent (5+sqrt(5))/4. Over larger alphabets, we give some candidates for balanced words (found computationally) having small critical exponents. We also explore a method for proving these results using the automated theorem prover Walnut.