Symmetric cocycles and classical exponential sums

This paper considers certain classical exponential sums as examples of cocycles with additional symmetries. Thus we simplify the proof of a result of Anderson and Pitt concerning the density of lacunary exponential partial sums $\sum_{k=0}^n exp(2πim^{k}x)$, n=1,2,..., for fixed integer m ≥ 2. Also, with the help of Hardy and Littlewood's approximate functional equation, but otherwise by elementary considerations, we improve a previous result of the author for certain examples of Weyl sum: if θ ∈ [0,1] \ ℚ has continued fraction representation $[a_{1},a_{2},... ]$ such that $\sum_{n} 1/a_{n} < ∞$, and $|θ - p/q| < 1/q^{4+ε}$ infinitely often for some ε $#62; 0, then, for Lebesgue almost all x ∈ [0,1], the partial sums $\sum_{k=0}^n exp(2πi(k^{2}θ + 2kx))$, n=1,2,..., are dense in ℂ.