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arxiv: 2605.01635 · v2 · submitted 2026-05-02 · 🧮 math.NT

A note on bilinear sums with modular square roots

Stephan Baier This is my paper

Pith reviewed 2026-05-08 19:30 UTC · model grok-4.3

classification 🧮 math.NT
keywords bilinear sumsexponential sumsmodular square rootsquadratic residuesanalytic number theoryfinite fieldsprime moduli
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The pith

Bilinear sums with modular square roots obey bounds analogous to the integer-exponent case.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that bilinear exponential sums of the form e_p(a x y^{1/2}), with y^{1/2} the modular square root of y modulo prime p when it exists, satisfy the same type of upper bounds previously obtained for integer exponents s. The proof adapts the method of Bag and Shparlinski with only routine changes to restrict y to quadratic residues inside the given interval. This matters for analytic number theory because such sums control the distribution of points or characters over finite fields. The note also reviews the author's recent related bilinear sum estimates.

Core claim

Closely following the method of Bag and Shparlinski, we establish an analogous result for the case when s=1/2 (y^{1/2} being a modular square root of y modulo p, if existent) for bilinear sums of terms of the form e_p(a x y^s), where p is a prime, a is coprime to p, x runs over a subset of F_p^*, and y runs over an interval.

What carries the argument

Adaptation of the Bag-Shparlinski bilinear sum technique to the square-root exponent, restricting the inner sum to quadratic residues modulo p.

If this is right

  • Upper bounds of the same shape as in the integer-s case now hold for these square-root bilinear sums.
  • The estimates apply whenever y is restricted to quadratic residues inside the interval.
  • The review of recent related bilinear sums places the s=1/2 case in a uniform context with prior integer-exponent results.
  • The same adaptation technique may be reused for other algebraic exponents whose images lie in a thin subset of F_p.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result suggests the original method is insensitive to replacing polynomial powers by algebraic functions whose range is a subgroup such as the quadratic residues.
  • It could be combined with character-sum techniques to bound the number of solutions to equations involving both linear and square-root terms modulo p.
  • Extending the same argument to higher roots or to sums over several variables would be a direct next step.

Load-bearing premise

The method of Bag and Shparlinski adapts directly to the square-root case with only routine modifications for the domain of quadratic residues.

What would settle it

A concrete counterexample: for a small prime p, explicit subset X of F_p^*, interval I, and a coprime to p, compute the sum over x in X and quadratic-residue y in I of e_p(a x y^{1/2}) and show its absolute value exceeds the claimed analogous bound by more than a fixed multiplicative constant.

read the original abstract

Bag and Shparlinski \cite{BaSh} considered bilinear sums of terms of the form $e_p(axy^s)$, where $p$ is a prime, $a$ is an integer coprime to $p$, $s$ is an integer, $x$ runs over a subset of $\mathbb{F}_p^{\ast}$ and $y$ runs over an interval. Closely following their method, we establish an analogous result for the case when $s=1/2$ ($y^{1/2}$ being a modular square root of $y$ modulo $p$, if existent). A part of this note is devoted to reviewing our recent works on related bilinear sums.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript extends the bilinear exponential sum estimates of Bag and Shparlinski for sums of the form e_p(a x y^s) to the case s = 1/2, where y^{1/2} denotes a chosen modular square root of y modulo a prime p (when it exists). The authors state that they obtain an analogous bound by closely following the original method, with routine modifications that restrict the y-sum to quadratic residues and select one branch of the square root. The note also reviews the authors' recent related results on bilinear sums.

Significance. If the claimed adaptation holds with a nontrivial bound, the result would furnish a useful estimate for exponential sums involving square roots modulo p. Such bounds can appear in applications to character sums over quadratic residues, distribution problems in finite fields, or sieve methods. The review of related works provides context but does not constitute the primary contribution.

major comments (1)
  1. [Abstract] Abstract: the claim that an 'analogous result' is established is not accompanied by an explicit statement of the bound or the error term obtained. Without this, it is impossible to verify whether the restriction to quadratic residues (which halves the range of y) preserves a nontrivial saving relative to the trivial bound or introduces a loss that renders the estimate weaker than the s-integer case in Bag-Shparlinski.
minor comments (1)
  1. The manuscript would benefit from a short paragraph clarifying how the original proof's treatment of the y-sum (interval versus residues) is modified when the support is restricted to quadratic residues; this would make the 'routine modifications' explicit rather than implicit.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater explicitness in the abstract. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that an 'analogous result' is established is not accompanied by an explicit statement of the bound or the error term obtained. Without this, it is impossible to verify whether the restriction to quadratic residues (which halves the range of y) preserves a nontrivial saving relative to the trivial bound or introduces a loss that renders the estimate weaker than the s-integer case in Bag-Shparlinski.

    Authors: We agree that the abstract would be improved by an explicit statement of the bound. The main theorem of the note establishes a bound of precisely the same form as the corresponding result in Bag and Shparlinski, with only routine modifications to restrict the inner sum to quadratic residues and to select one branch of the square root. Because the original estimates depend on the length of the y-interval rather than on the precise density of the summation set, the factor of roughly 1/2 arising from the quadratic residues is absorbed into the implied constants and does not diminish the saving relative to the trivial bound. We will revise the abstract to state the bound explicitly (quoting the statement of the main theorem) and to note that the saving is preserved. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows external method

full rationale

The paper's central result is an adaptation of the bilinear sum estimates from the externally cited work of Bag and Shparlinski (BaSh), restricted to the case s=1/2 by restricting the y-sum to quadratic residues and selecting a branch of the square root. This is presented as a direct, routine extension without introducing new fitted parameters or self-defined quantities that would force the bound by construction. The mention of reviewing the author's own recent works on related sums is supplementary and does not serve as the load-bearing justification for the main theorem; the proof strategy is anchored in the cited external technique rather than reducing to prior self-citations or ansatzes. No equations or steps equate a claimed prediction to an input defined within this paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard facts about additive characters over finite fields and the definition of modular square roots on quadratic residues; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Properties of the additive character e_p(z) = exp(2π i z / p) on F_p
    Used throughout the definition of the bilinear sums.
  • domain assumption Square roots exist precisely when the argument is a quadratic residue modulo p
    Implicit in restricting the sum to y where y^{1/2} is defined.

pith-pipeline@v0.9.0 · 5401 in / 1246 out tokens · 45931 ms · 2026-05-08T19:30:38.908474+00:00 · methodology

discussion (0)

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Reference graph

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