A twisted Kloosterman sum is a character sum of the form $$S(\chi, \psi, \eta)=\sum_{t\in (\mathbb{F}_q)^{\times}} \chi(t) \psi(t) \eta(t^{-1}).$$ Here, $\chi$ is a multiplitive character of $(\mathbb{F}_q)^{\times}$ and $\psi, \eta $ are additive characters. I have two questions:

  1. Why were people so interested in bounding these types of character sums? For example, I know a lot of work went in to actually proving the bound for $|S(\chi, \psi, \eta)|$.

  2. Do sums of twisted Kloosterman sums appear in any relevant context? For instance a sum $$\sum_{\chi\in \widehat{\mathbb{F}_q^{\ast}}}S(\chi, \psi, \eta).$$

  • $\begingroup$ According to Lidl&Niederreiter Davenport was the first to study these sums. I do not know what motivated him. Chowla was the first to prove the Weil type bound, when $\chi^2\not\equiv1$. I have seen it used (and have used related sums myself) in a coding theoretical setting. If you build families of e.g. signature or scrambling sequences as products like $s(i)=\psi(g^t)\eta(g^{-t})$, where $g$ is a primitive element (=a generator of the multiplicative group), then you need these twisted sums $\endgroup$ – Jyrki Lahtonen Jan 28 '16 at 13:19
  • $\begingroup$ (cont'd) (L&N use the term generalized Kloosterman sums), then you can derive bounds for the so called partial period correlation functions of those sequences using the (Polya-)Vinogradov method, if you have a good bound for the generalized Kloosterman sums. Vinogradov method is much older, but I take some of the credit for popularizing its use in coding theory. Basically because I called it (in a paper from '95) what the engineers were familiar with: A Discrete Fourier Transform :-) $\endgroup$ – Jyrki Lahtonen Jan 28 '16 at 13:23

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