There's an idea used in the analysis of error correcting codes, to set up a polynomial where the coefficients are the number of codewords with each Hamming weight. It works out the same as adding up all the codewords, where you replace each 0 by x and each 1 by y. So a code
010 101 => xyx + yxy = x^2y + xy^2
Then you can use real (or even complex) analysis on these polynomials. There's a thick book of things you can prove about sets of binary words using those techniques: https://www.elsevier.com/books/the-theory-of-error-correctin...
And Noam Elkies' course notes on it: http://www.math.harvard.edu/~elkies/M256.13/index.html