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 ${\displaystyle \sum _{k=0}^n\exp (2\pi i{m}^{k}x)}$, 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 ${\displaystyle [a_1,a_2,...]}$ such that ${\displaystyle \sum _n\frac{1}{a_n}<\infty }$, and ${\displaystyle |\theta -\frac{p}{q}|<\frac{1}{q^{4+\epsilon }}}$ infinitely often for some ε ${\displaystyle \#62;0,then,f\text{or}Lebesguealmostallx\in [0,1],the\partial \sum s}$\sum_{k=0}^n exp(2πi(k^{2}θ + 2kx))!$!, n=1,2,..., are dense in ℂ.