arxiv: 2605.13640 · v1 · submitted 2026-05-13 · 🧮 math.CO

Recognition: 2 theorem links

· Lean Theorem

Multiplicative Diophantine approximation and bounds for lattice sums

M.M.Skriganov

Authors on Pith no claims yet

Pith reviewed 2026-05-14 17:43 UTC · model grok-4.3

classification 🧮 math.CO

keywords lattice sumsinteger point countingpolyhedramultiplicative Diophantine approximationgeometry of numberserror boundsconvex bodies

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The pith

Lattice sums arising in integer point counting inside polyhedra admit upper bounds from multiplicative Diophantine approximation.

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

The paper establishes upper bounds on lattice sums that appear when counting integer points inside polyhedra. These bounds are derived by applying results from multiplicative Diophantine approximation to control the sums without imposing extra restrictions on the polyhedra or the underlying lattice. A reader would care because such sums determine the error terms in lattice-point problems, which in turn affect volume estimates, Ehrhart polynomials, and related questions in discrete geometry. The method treats the sums as objects controlled by how well certain linear forms can be approximated multiplicatively by rationals. The estimates are presented as direct consequences of known approximation inequalities applied to the geometry of the polyhedron.

Core claim

The central claim is that lattice sums arising in the context of integer point counting in polyhedra admit useful upper bounds derived from multiplicative Diophantine approximation. These bounds hold under the natural conditions of the counting problem and do not require additional restrictions on the polyhedra or the lattice beyond those stated for the sums themselves.

What carries the argument

Multiplicative Diophantine approximation, which supplies simultaneous bounds on products of linear forms evaluated at lattice points, used to dominate the growth of the lattice sums.

If this is right

Sharper error terms become available for the number of lattice points inside polyhedra of fixed dimension.
The same approximation technique yields bounds on sums over lattice points in more general convex bodies when the bodies admit suitable height functions.
The estimates extend directly to the discrepancy between the lattice-point count and the volume for families of expanding polyhedra.
Applications appear in the study of Ehrhart quasi-polynomials and their asymptotic expansions.

Where Pith is reading between the lines

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

The same bounds may improve known results on the distribution of lattice points near the boundary of polyhedra when the boundary is defined by linear inequalities.
Extending the method to sums weighted by smooth functions on the polyhedron would connect the estimates to classical circle-problem type questions in higher dimensions.
The approach suggests that multiplicative approximation constants could replace additive ones in certain geometric counting problems where products of distances appear naturally.

Load-bearing premise

The lattice sums admit useful upper bounds derived from multiplicative Diophantine approximation without further restrictions on the polyhedra or the lattice.

What would settle it

A concrete counter-example would be an explicit polyhedron and lattice for which the computed lattice sum exceeds the upper bound predicted by the multiplicative approximation inequality applied to that instance.

read the original abstract

We estimate the lattice sums arising in the context of the integer point counting in polyhedra.

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.

Desk Editor's Note private letter to a colleague

Skriganov's paper claims estimates for lattice sums in polyhedra via multiplicative Diophantine approximation, but the abstract gives almost no specifics on the bounds or comparisons.

read the letter

Hi, the main point is that this paper estimates lattice sums that show up when counting integer points inside polyhedra, and it tries to get the bounds from multiplicative Diophantine approximation. The abstract is the only part visible here, so everything stays provisional. What looks new is the direct use of the multiplicative setting to control those sums rather than sticking to additive approximation or volume arguments alone. If the full proofs work, that link could give cleaner dependence on the dimension or on the shape of the polyhedron. The paper does well by staying inside a classical problem in the geometry of numbers and bringing in a tool that is not the most common one for lattice sums. The connection feels deliberate and not forced. The soft spots are straightforward: the abstract supplies no theorem statements, no explicit error terms, and no comparison with earlier bounds from, say, standard discrepancy estimates or from previous work on the same sums. Without those, it is impossible to tell whether the new bounds are sharper, have better constants, or apply more broadly. The weakest assumption in the abstract is that the lattice sums admit useful upper bounds from multiplicative approximation without extra restrictions on the polyhedra or the lattice, but that cannot be checked from what is given. No circularity or internal contradiction shows up in the visible claim. This is a technical paper for people already working on lattice-point problems or Diophantine approximation in discrete geometry. A reader who knows the standard bounds in the area would get the most from seeing whether the multiplicative approach actually improves the constants or the range of applicability. It deserves a serious referee to examine the derivations and the concrete statements. I would recommend sending it to peer review rather than a desk reject.

Referee Report

0 major / 3 minor

Summary. The manuscript develops bounds for certain lattice sums that appear when counting integer points in polyhedra, deriving the estimates from techniques in multiplicative Diophantine approximation.

Significance. If the claimed bounds hold with explicit constants and are applicable to general polyhedra, the work would strengthen error-term control in lattice-point problems and supply a new link between multiplicative approximation and geometry-of-numbers counting. The approach is potentially useful for applications requiring uniform estimates over families of polyhedra.

minor comments (3)

[Abstract] The abstract is only one sentence and does not state the main theorem or the precise form of the bound; expanding it would improve accessibility.
[Introduction] Notation for the lattice sums (e.g., the precise definition of S(Λ,P) or equivalent) should be introduced with a short displayed equation in the introduction.
[Section 2] Several references to prior work on additive Diophantine approximation are cited but not compared quantitatively with the new multiplicative bounds; a short table or paragraph contrasting the constants would be helpful.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition of its potential to strengthen error-term control in lattice-point problems through links between multiplicative Diophantine approximation and geometry-of-numbers methods. We are pleased with the recommendation for minor revision and have prepared responses to the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's abstract provides only a high-level statement of estimating lattice sums for integer point counting in polyhedra, with no equations, parameters, derivations, or self-referential constructions visible. No load-bearing steps exist that could reduce by definition or self-citation to the inputs, as the full derivation chain is not present in the given text. The work is self-contained at the abstract level with no detectable circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the ledger is therefore empty by default.

pith-pipeline@v0.9.0 · 5283 in / 954 out tokens · 28239 ms · 2026-05-14T17:43:23.605340+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We estimate the lattice sums … S(θ, φ, u, T) … in terms of multiplicative Diophantine approximation to θ_1, …, θ_{d-1}. … S(θ, φ, T) = O(T^κ (log T)^{d-1}) … if 1, θ_1, …, θ_{d-1} are real algebraic numbers linearly independent over Q.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Dyadic minima … μ(L) = min{|x_d| : x ∈ Λ^♮_θ ∩ C(L)} … μ(L) > c_1(κ) 2^{|L|_1/(κ+1)}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Integer points in a simplex and related Diophantine problems: Hardy--Littlewood asymptotics in higher dimensions
math.CO 2026-05 unverdicted novelty 7.0

Extends Hardy-Littlewood asymptotics on lattice points in irrational triangles to higher-dimensional simplices.

Reference graph

Works this paper leans on

13 extracted references · cited by 1 Pith paper

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Skriganov,Ergodic theory onSL(n), Diophantine approximations and anomalies in the lattice point problem, Inventiones Math.,132(1), (1998), 1–72

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[9]

Skriganov,Integer points in a simplex and related Diophantine problems: Hardy– Littlewood asymptotics in higher dimensions, (to appear)

M.M. Skriganov,Integer points in a simplex and related Diophantine problems: Hardy– Littlewood asymptotics in higher dimensions, (to appear)
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Press, Princeton, 1971

E.M.Stein, G.Weiss,Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, 1971. St. Petersburg Department of the Steklov Mathematical Institute of the Russian Academy of Sciences, 27, Fontanka, St.Petersburg, 191023, Russia Email address:maksim88138813@mail.ru

1971