## Abstract

We prove nontrivial estimates for linear sums of squares of Fourier coefficients of holomorphic and Maass cusp forms twisted by additive characters. For holomorphic forms *f*, we show that if \(|\alpha -a/q| \le 1/q^2\) with \((a,q)=1\), then for any \(\varepsilon >0\),

Moreover, for any \(\varepsilon > 0,\) there exists a set \(S \subset (0, 1)\) with \(\mu (S)=1\) such that for every \(\alpha \in S\), there exists \(X_0=X_0(\alpha )\) such that the above inequality holds true for any \(\alpha \in S\) and \(X \geqslant X_0(\alpha ).\) A weaker bound for Maass cusp forms is also established.

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## Acknowledgements

The main part of this work was done during the author’s Ph.D. thesis at the Indian Statistical Institute (ISI), Kolkata where he was a student. The author thanks Satadal Ganguly, Stephan Baier and Ritabrata Munshi for helpful suggestions. The author also thanks ISI for financial support and excellent working atmosphere.

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Communicating Editor: A Raghuram

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Acharya, R. Exponential sums of squares of Fourier coefficients of cusp forms.
*Proc Math Sci* **130, **24 (2020). https://doi.org/10.1007/s12044-019-0550-4

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### Keywords

- Cusp forms
- exponential sums
- diophantine approximation

### Mathematics Subject Classification

- 11F30