Type-II Shifted Bilinear Sums

• Type-II shifted bilinear sums are double sums over long sequences with nonlinear, shifted arithmetic functions that are vital for analysis in number theory.
• They are examined using techniques such as the circle method, amplification, and Voronoi summation to achieve breakthrough bounds in shifted convolution and subconvexity problems.
• Applications include improved estimates for L-functions, refined error terms in shifted convolutions, and progress in areas like automorphic forms and additive combinatorics.

A type-II shifted bilinear sum refers, in analytic number theory, to a bilinear form involving two (typically independent) sequences, both sufficiently long, with a “shifted” or nonlinear arithmetic argument. These sums are of fundamental importance due to their ubiquity in the analysis of shifted convolution sums, subconvexity of $L$-functions, and moments of Dirichlet polynomials. Their structure, analysis, and non-trivial estimation constitute a central theme in automorphic forms, additive combinatorics, and the analytic theory of $L$-functions.

1. Definition and General Structure

A general type-II shifted bilinear sum is a double sum of the form

$$\sum_{m \sim M} \sum_{n \sim N} \alpha_m \beta_n\ F(m, n)$$

where $F$ is an arithmetic (often oscillatory) function exhibiting nonlinear, shifted, or correlated dependence on $m$ and $n$, and both $M$ and $N$ are “long” ranges (e.g., $M,N \ge X^{\delta}$ for some $\delta > 0$).

Key examples include:

• Sums of the form $\sum_{m, n} \alpha_m \beta_n S(a m, b n + r; p)$ with $S(x, y; p)$ a Kloosterman sum and arbitrary or shifted arguments (Kerr et al., 2022).
• Sums with $F(m, n) = \mathrm{Kl}_k(\pm c m n+h; q)$, the normalized $k$-hyper-Kloosterman sum evaluated at an affine or shifted argument (Kowalski et al., 2015).
• Shifted convolution sums, e.g., $\sum_{n} a(n) b(n+h)$, which upon analytic decoupling (e.g., via the delta/DFI method or spectral methods) yield type-II bilinear pieces (Harun et al., 7 Jan 2025, Molla, 28 Nov 2025, Conrey et al., 2016, Leung, 2022, Munshi, 2012).
• Bilinear forms with trace functions or general $\ell$-adic sheaf trace weights over finite fields with shifted arguments (Fouvry et al., 12 Nov 2025).

Type-II sums are distinguished from Type-I sums (where one sequence is taken to be extremely short, or one variable is fixed/weighted trivially) by the presence of two genuinely long variables.

2. Analytic Framework and Decomposition

Type-II shifted bilinear sums emerge as crucial components in the analytic decomposition of shifted convolutions and moments. Typical frameworks yielding such sums include:

• Circle Method / Delta Method: The insertion of an analytic delta symbol (as in Duke–Friedlander–Iwaniec or Jutila’s circle method) to detect equations like $m + h = n$, followed by dualizing via Poisson and Voronoi summation, systematically generates bilinear sums in “dual” variables (Harun et al., 7 Jan 2025, Kaneko, 2023, Conrey et al., 2016, Munshi, 2012).
• Amplification and Divisor-Switching: In moments of $L$-functions, amplification or the delta method singles out shifted bilinear “off-diagonal” terms, often after shift-splitting or divisor switching (Conrey et al., 2016, Kaneko, 2023).
• Spectral or Trace-Function Decomposition: Bilinear forms with trace functions over finite fields, or in the context of automorphic forms, generate shifted or nonlinear bilinear forms after decoupling via spectral tools or stratification arguments (Fouvry et al., 12 Nov 2025, Diamantis et al., 2013).

In all cases, careful partitioning distinguishes the main diagonal, Type-I, and Type-II contributions, with the latter containing the deepest analytic difficulty and often forming the bottleneck for subconvexity or mean value results.

3. Principal Results and Best-Known Bounds

Recent developments have produced sharp “barrier-beating” bounds for a wide variety of Type-II shifted bilinear sums. Key results include:

Sum Type / Reference	Canonical Form	Main Bound / Saving
Kloosterman sum, prime mod (Kerr et al., 2022)	$\sum \alpha_m \beta_n S(a m, b n + r; p)$	$$|B| \ll p^{\varepsilon} \|\alpha\|_2 \|\beta\|_\infty M^{1/2}N \left( \frac{p}{MN} \right)^{1/4}$$; nontrivial as soon as $MN > p^{\delta}$
Hyper-Kloosterman (Kowalski et al., 2015)	$$\sum \alpha_m \beta_n \mathrm{Kl}_k(\pm c m n + h; q)$$	$$|B| \le C_{k,\epsilon}|c|q^{\epsilon}\|\alpha\|_2\|\beta\|_2 [(MN)(M^{-1} + (MN)^{-1/6}q^{1/2})]^{1/2}$$; saving of $q^{-1/64+o(1)}$ at $M=N=q^{1/2}$
Trace functions, general (Fouvry et al., 12 Nov 2025)	$\sum \alpha_m \beta_n K(am+bn+h)$ with $K$ a gallant sheaf	$$\ll q^\varepsilon \|\alpha\|_2\|\beta\|_2(MN)^{1/2-\eta}$$ for $MN < q^{3/4+\delta}$
GL(3) × GL(3) shifted convolution (Harun et al., 7 Jan 2025)	$\sum_h \sum_n A_{T_1}(1, n) A_{T_2}(1, n+h)$	$L(H, X) \ll_{\text{forms}} X^{1-\delta + \epsilon}$ for $H > X^{1/2 + \delta}$
Kloosterman fractions (Dong et al., 1 Jan 2026)	$\sum \alpha_m \beta_n e(a \overline{m} / (b n))$	$$\ll \|\alpha\|_2 \|\beta\|_2 (M+N)^{1/6} \min\{M,N\}^{1/3} \max\{M,N\}^{-1/12+\varepsilon}$$

These results have systematically improved the power-saving exponents accessible in analytic applications such as mean values of $L$-functions, breaking the Pólya–Vinogradov $N \asymp q^{1/2}$ barrier, and allowing for uniform savings in various nontrivial regimes.

4. Methodologies and Key Techniques

Nontrivial bounds for Type-II shifted bilinear sums typically exploit a combination of analytic and algebraic methods:

• Analytic Partitioning (Cauchy–Schwarz, Delta Symbol, Poisson/Voronoi Summation):

Decoupling steps yield sums of products, allowing use of $L^2$-norms and spectral tools. In the context of $GL(3) \times GL(3)$ or $GL(d)$ convolution, iterated Voronoi summation is standard (Harun et al., 7 Jan 2025, Molla, 28 Nov 2025, Munshi, 2012).

• Amplification and Input-Averaging:

Employing short amplifiers, often in one variable, increases the length of “diagonal” or nearly diagonal sums, against which the “off-diagonal” (genuinely shifted) part is measured (Dong et al., 1 Jan 2026, Pascadi, 11 Nov 2025).

• Non-abelian Harmonic Analysis:

For composite moduli (e.g. products of two primes), advanced representation theory and Fourier analysis on groups such as $\mathrm{SL}_2(\mathbb{Z}/c\mathbb{Z})$ permit uniform savings for all moduli and push exponents down to the theoretical limit for currently available techniques (Pascadi, 11 Nov 2025).

• Stratification and Monodromy Arguments:

In the context of trace functions, Junyan Xu’s “soft stratification” and group-theoretic Goursat-Kolchin-Ribet criteria reduce the analysis to diagonal loci and allow square-root saving by Deligne’s Riemann Hypothesis (Fouvry et al., 12 Nov 2025, Kowalski et al., 2015).

• Geometric Tools (Deligne’s Theorem, $\ell$-adic sheaf theory):

Complete exponential sums in more than two variables, which arise naturally after applying duality and Poisson, are analyzed using algebraic geometry over finite fields, pure $t$-motives, and monodromy group computations (Kowalski et al., 2015, Fouvry et al., 12 Nov 2025).

5. Applications and Impact

Type-II shifted bilinear bounds have found application in a wide range of problems:

• Moments and Subconvexity: Controlling mean values and moments of automorphic $L$-functions in various aspects ($t$-aspect, level-aspect, twist-aspect) (Conrey et al., 2016, Kaneko, 2023, Molla, 28 Nov 2025).
• Distribution in Arithmetic Progressions: Power-savings in divisor function and eigenvalue distributions in arithmetic progressions to large moduli (Kerr et al., 2022).
• Spectral Theory and $L^2$ Restrictions: Estimates for spectral sums and shifted convolutions on higher rank groups (Harun et al., 7 Jan 2025, Leung, 2022).
• Mean Square Error Terms: Mean square estimates for Dirichlet polynomials and error terms in the asymptotics of shifted convolution sums, arising in the context of the Riemann zeta function (Conrey et al., 2016).
• Ramanujan Expansions and Finite Expansion Methods: Type-II sums explicitly arise in the off-diagonal terms when using finite Ramanujan expansions for heuristic shifted convolution sum evaluation (Coppola et al., 2017).
• Salié Sums and Modular Square Roots: Distribution of quadratic residues over shifted primes critically depends on Type-II estimates involving Salié sums (Shparlinski et al., 15 Jan 2026).

6. Connections to Broader Theories and Open Directions

Type-II shifted bilinear sums act as a “gateway” problem with implications extending into combinatorial incidence theory, random matrix analogies, and automorphic representation theory. Open challenges and directions include:

• High Rank and Lindelöf Range: For $GL(d)\times GL(d)$, breaching the threshold $H \leq N^{1-2/d}$ in shifted convolution would yield new subconvexity exponents (Molla, 28 Nov 2025).
• Uniformity Over All Moduli: Combining non-abelian amplification, sum-product methods, and harmonic analysis may yield ultimate uniformity in Kloosterman and trace-weighted sums (Pascadi, 11 Nov 2025, Kerr et al., 2022).
• Saturated Case and Beyond: Extraction of further savings in the “critical” range (i.e., at the threshold of current methods) is closely tied to deep monodromy, irreducibility, and entropy phenomena in the underlying sums (Kowalski et al., 2015, Fouvry et al., 12 Nov 2025).
• Hypergeometric and Trace Function Generality: Extending the existing techniques to arbitrary trace functions of higher complexity and large conductor with shifted arguments is a frontier under active development (Fouvry et al., 12 Nov 2025).

7. Exemplary Results and Comparative Table

Reference	Sum Type	Variable Lengths	Power Saving
(Dong et al., 1 Jan 2026)	Kloosterman fractions	$M=N$	$1/12$ (best known for general coefficients)
(Kowalski et al., 2015)	Hyper-Kloosterman, shifted	$M=N \sim q^{1/2}$	$q^{-1/64+o(1)}$
(Fouvry et al., 12 Nov 2025)	General trace functions	$M N \leq q^{3/4+\delta}$	$(MN)^{-\eta}$ for some $\eta>0$
(Harun et al., 7 Jan 2025)	$GL(3) \times GL(3)$ shifted conv.	$H > X^{1/2+\delta}$	$X^{-\delta}$ for any $\delta>0$
(Pascadi, 11 Nov 2025)	Kloosterman sums, all moduli	$M=N\sim \sqrt{c}$	$c^{-1/12}$

These nontrivial savings underpin many modern advances in analytic number theory and the study of automorphic forms.

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References:

• (Harun et al., 7 Jan 2025) Shifted convolution sum with weighted average: $GL(3) \times GL(3)$ setup
• (Molla, 28 Nov 2025) Average Shifted Convolution sum for $GL(d_1)\times GL(d_2)$
• (Kowalski et al., 2015) Bilinear forms with Kloosterman sums and applications
• (Kerr et al., 2022) Bounds on bilinear forms with Kloosterman sums
• (Fouvry et al., 12 Nov 2025) Bilinear forms with trace functions
• (Coppola et al., 2017) Finite Ramanujan expansions and shifted convolution sums II
• (Dong et al., 1 Jan 2026) Bilinear forms with Kloosterman fractions and applications
• (Munshi, 2012) Shifted convolution sums for $GL(3)\times GL(2)$
• (Leung, 2022) Shifted Convolution Sums for $GL(3)\times GL(2)$ Averaged over weighted sets
• (Conrey et al., 2016) Moments of zeta and correlations of divisor-sums: IV
• (Kaneko, 2023) On Multiple Shifted Convolution Sums
• (Shparlinski et al., 15 Jan 2026) Shifted bilinear sums of Salié sums and the distribution of modular square roots of shifted primes
• (Kajihara, 2013) Transformation formulas for bilinear sums of basic hypergeometric series