This work presents a study of perturbations of symmetric Boolean functions. In particular, it establishes a connection between exponential sums of these perturbations and Diophantine equations
of the form
$$ \sum_{l=0}^n \binom{n}{l} x_l=0,$$
where \(x_l\) belongs to some fixed bounded subset \(\Gamma\) of \(\mathbb{Z}\). The concepts of trivially balanced symmetric Boolean function and sporadic balanced are extended to this type of perturbations.
An observation made by Canteaut and Videau for symmetric Boolean functions of fixed degree is extended. To be specific, it is proved that, excluding the trivial cases,
balanced perturbations of fixed degree do not exist when the number of variables grows. Some sporadic balanced perturbations are presented. Finally, a beautiful but unexpected identity between
perturbations of two very different symmetric Boolean functions is also included in this work.
