On certain bilinear sums with modular square roots and applications

We extend bounds on additive energies of modular square roots by Dunn, Kerr, Shparlinski, Shkredov and Zaharescu and apply these results to obtain bounds on certain bilinear exponential sums with modular square roots. From here, we make partial progress on the large sieve for square moduli.

💡 Research Summary

The paper investigates additive energies associated with modular square roots and applies the resulting bounds to obtain non‑trivial estimates for certain bilinear exponential sums, ultimately yielding conditional progress on the large sieve for square moduli.

The author begins by recalling the state‑of‑the‑art bounds for the additive energies (T_2(r,j,M)) and (T_4(r,j,M)) established by Dunn, Kerr, Shparlinski, Shkredov and Zaharescu. These energies count quadruples (or octuples) of integers whose modular square‑root differences satisfy specific congruences. The novelty of this work lies in introducing a “distance restriction’’ parameter (H) and defining restricted energies (E_2(r,j,M,H)) and (E_4(r,j,M,H)) where the differences (|m_i-m_{i+1}|) are bounded by (H). By translating the congruence conditions into lattice point problems, the author applies Minkowski’s second theorem and the Betke–Henk–Wills bound on the number of lattice points in a convex body. This yields the unconditional estimates

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