Gaussian dyadic summation bound (${\sum }_{j\ge 0}{e}^{\epsilon jh-c(jh{)}^2/L}{\ll }_{c,h}\sqrt{L}{e}^{{\epsilon }^{2}L/(4c)}$)

For fixed $c,h>0$, $L\ge 1$ and $\epsilon \ge 0$, complete the square in the exponent. A translated Gaussian function summed on a fixed-spaced lattice has mass $O(\sqrt{L})$ uniformly in the translate, by comparison with its integral and its maximum. This keeps a square-root logarithm, rather than the full number of dyadic intervals, when summing exponentially damped exponential sum bounds.