[論文レビュー] On certain bilinear sums with modular square roots and applications

Stephan Baier|arXiv (Cornell University)|Jan 21, 2026

Analytic Number Theory Research被引用数 0

ひとこと要約

tldr: 本論文はモジュラー平方根の可換的和に関する加法エネルギーの境界を拡張し、それをモジュラー平方根を含む二項指数和を境界付けるのに用い、平方模数に対する大いすう sieve に関して部分的な進展を示す。

ABSTRACT

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.

研究の動機と目的

• Extend bounds on additive energies of modular square roots as in prior work by Dunn, Kerr, Shparlinski, Shkredov and Zaharescu.
• Apply these energy bounds to obtain non-trivial bounds on bilinear exponential sums with modular square roots.
• Use the results to derive partial progress on the large sieve for square moduli.

提案手法

• Develop and bound the bilinear sums with modular square roots of the form Sigma(r,j,L,M,alpha,beta,f).
• Employ additive energy analysis via lattice geometry, Minkowski’s theorem and Betke–Hank–Wills type lattice counting.
• Transform algebraically the square-root congruences to tractable counting problems (via m1,m2, h, d variables) and partition sums by lattice minima.
• Prove bounds (Theorem 2) for general oscillatory sums involving modular square roots and a differentiable analytic phase f.
• Derive energy bounds E2 and E4 (Theorems 4) and connect these to bilinear sum bounds (Theorem 5) and large-sieve implications (Theorem 6).

実験結果

リサーチクエスチョン

• RQ1 What non-trivial bounds can be established for bilinear sums involving modular square roots when the bilinear ranges are smaller than sqrt(r)?
• RQ2 How do refined additive-energy bounds of modular square roots influence bilinear exponential sums and large sieve estimates for square moduli?
• RQ3 Under what hypothesis on E4(r,j,M,H) can one obtain improved large-sieve at the critical point N=Q^3?
• RQ4 How can lattice-geometry methods be employed to bound restricted energies E2 and E4?
• RQ5 What conditional improvements to the large sieve for square moduli follow from Hypothesis 1 and Theorem 5?

主な発見

• Theorem 2 provides non-trivial bounds for the bilinear sums Sigma(r,j,L,M,alpha,beta,f) under constraints on L, M, F, and H.
• Corollary 1 yields a non-trivial bound for Sigma(r,j,L,M,alpha,beta) in a regime where L and M are up to r and M is between r^{1/3} and r, improving over trivial bounds in certain ranges.
• Theorem 4 gives energy bounds E2(r,j,M,H) and E4(r,j,M,H) with explicit dependence on H, M, and r, under general moduli r and gcd conditions.
• Theorem 5 introduces a weaker yet usable hypothesis (Hypothesis 1) on E4(r,j,M,H) that leads to improved bilinear-sum bounds (Theorem 5) and corollaries.
• Corollary 2 translates Hypothesis 1 into a concrete bound for Sigma(r,j,L,M,alpha,beta) in terms of L, M, r, and nu, enabling conditional progress.
• Theorem 6 shows a conditional improvement of the large sieve for square moduli at the critical point N=Q^3 under Hypothesis 1.