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An arithmetic measure of width for convex bodies

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Pith's one-line read A new invariant, arithmetic width, counts the distinct values a linear functional attains on the lattice points of a convex body, refining lattice width and enabling fixed-dimension computation.

desk verdict New invariant, solid structural theorems, but Main Theorem 3 has two repairable proof gaps that need fixing before the algorithmic claims are accepted. read the letter →

arxiv 2509.04726 v1 pith:RPXJZWJN submitted 2025-09-05 math.CO

Jesús A. De Loera , Brittney Marsters , Christopher O'Neill This is my paper · ORCID
classification math.CO MSC 11H0652B2052C07
keywords arithmeticwidthlatticeconvexbodyrationalpolytopeprogressionquasilineargrowthgeneratingfunctionsnumericalsemigroup
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces the arithmetic width of a convex body, defined for a fixed integer direction as the number of distinct values a linear functional takes on the lattice points inside the body. Unlike lattice width, which measures distances between parallel supporting hyperplanes, arithmetic width counts only the hyperplane levels that actually contain lattice points, so gaps in the lattice-point distribution are detected. The central claims are structural and algorithmic: for large dilates of any convex body the attained values form an arithmetic progression with only bounded omissions near the ends, and for rational polytopes the arithmetic width, whether in a fixed direction or minimized over all directions, grows eventually quasilinearly in the dilation parameter (linear on each residue class modulo the denominator) while the optimal directions repeat periodically. In fixed dimension, the arithmetic width in a given direction is computable in polynomial time, and the global arithmetic width is computable in single-exponential time, using short rational generating functions and a finite test set of directions.

What carries the argument

The central object is the arithmetic width $\operatorname{aw}_c(K)=|\mathrm{AR}_c(K)|$, the number of distinct values of $c^{\mathsf{T}}x$ on $K\cap\mathbb{Z}^d$. The structure theorem is carried by a convexity argument: once a dilation is large enough to contain a small ball, the fact that a primitive integer direction attains every integer value on the lattice guarantees that every intermediate lattice hyperplane contains a lattice point, so the arithmetic range is an arithmetic progression except near the extremes. The algorithmic machinery is the short rational generating function of the lattice points of a rational polytope, a polynomial-size expression in fixed dimension, together with substitution and projection theorems that turn it into a univariate generating function whose distinct monomials count the arithmetic width. For the global width, the finite test set $T$ is built from primitive integer vectors orthogonal to linearly independent differences of lattice points of $P$, since any minimizing direction must collapse two distinct lattice points.

What would settle it

For the simplex $P=\operatorname{conv}((0,0,0),(2,0,0),(1,1/3,1/3),(1,-1/3,1/3))$ in $\mathbb{R}^3$, the nonzero integer differences $(P-P)\cap\mathbb{Z}^3$ are all multiples of $(1,0,0)$, so the enumeration in Lemma 5.4 returns no test directions; yet $\operatorname{aw}(P)=1$, attained by any direction perpendicular to the $x$-axis. Running the stated algorithm on this input would miss the optimum, showing that the constructed set is not a sufficient test set in the form presented.

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Extended reading notes

Core claim

The core discovery is that the arithmetic range $\mathrm{AR}_c(K)=\{c^{\mathsf{T}}x : x\in K\cap\mathbb{Z}^d\}$ of a convex body has a stabilized, almost arithmetic shape under dilation. For large $n$, the set $\mathrm{AR}_c(nK)$ agrees with an arithmetic progression of some step $\lambda$ except for a bounded number of missing values near the minimum and maximum, where $t,t',\lambda$ depend only on $K$. For a rational polytope $P$ with denominator $D$, the arithmetic width in a fixed direction is eventually quasilinear in $n$, and the same holds for the minimized width $\operatorname{aw}(nP)=\min_c \operatorname{aw}_c(nP)$; moreover the minimizing directions recur periodically with period $D$. On the algorithmic side, the arithmetic width in a fixed direction is computable in polynomial time in fixed dimension, and the global arithmetic width is computable in single-exponential time via a finite test set of directions obtained from differences of lattice points. Finally, the optimal directions for arithmetic and lattice width can be disjoint, and every finite set of integers arises as the arithmetic range of a rational simplex.

Load-bearing premise

The test-set construction assumes that the differences of lattice points of $P$ contain enough linearly independent vectors to determine every minimizing direction; a full-dimensional rational polytope whose lattice points all lie on a line makes the enumeration come up empty even though minimizing directions exist.

Editorial extensions

If this is right

  • For any rational polytope $P$ with denominator $D$, the arithmetic width $\operatorname{aw}(nP)$ is eventually quasilinear in $n$, so the width of dilates is described by $D$ linear functions, one per residue class modulo $D$, and the optimal directions repeat with period $D$.
  • In fixed dimension, given any integer direction $c$, $\operatorname{aw}_c(P)$ can be computed in polynomial time in the bit-sizes of $P$ and $c$, and $\operatorname{aw}(P)$ can be computed in single-exponential time.
  • Arithmetic width is at most lattice width plus one and detects gaps that lattice width ignores; Example 2.2 shows the directions minimizing the two invariants can be disjoint, so arithmetic width is not just a relabeling of lattice width.
  • Because every finite set of integers is realizable as the arithmetic range of some rational simplex, no simple structural restriction other than finiteness governs which sets can be arithmetic ranges; in particular the realization result for numerical semigroup factorization lengths follows as a corollary.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The almost-arithmetic-progression structure of arithmetic ranges parallels the eventual quasipolynomial behavior of lattice-point counting functions of rational polytopes and of graded algebra dimension sequences, so arithmetic width could serve as a discrete-geometric analog of those counting functions.
  • Because $\operatorname{aw}(P)$ counts occupied hyperplane levels, it can serve as a sharper certificate of thinness in integer programming: a small arithmetic width in a near-optimal direction indicates that few integer objective values are available, complementing existing integrality-gap estimates.
  • A natural next step is to ask whether the finite test set can be constructed from the lattice span of $(P-P)\cap\mathbb{Z}^d$ rather than from the full difference set; answering this would settle the algorithm's scope for polytopes whose lattice points occupy a lower-dimensional subspace.
  • The realization theorem suggests a construction tool: one can design rational simplices whose lattice-point projections onto a line realize any prescribed finite pattern, which may be useful for building examples in projection and lifting problems.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper introduces the arithmetic width of a convex body K, defined as the minimum over nonzero integer directions c of the number of distinct values that c^T x attains on the lattice points K∩Z^d. The authors establish that for large dilates nK, the arithmetic range in a fixed direction is an almost arithmetic progression (Main Theorem 1); that for rational polytopes the arithmetic width in a fixed direction, and the minimized arithmetic width, are eventually quasilinear in the dilation parameter (Main Theorem 2); and that in fixed dimension, aw_c(P) is computable in polynomial time and aw(P) in single-exponential time (Main Theorem 3). They also show that optimal directions for lattice width and arithmetic width can differ, and that every finite set of integers occurs as an arithmetic range of a rational simplex (Main Theorem 4).

Significance. The notion of arithmetic width is a natural refinement of lattice width, and the structural results linking it to almost arithmetic progressions and eventual quasilinearity are genuinely interesting, connecting discrete geometry, integer programming, and additive combinatorics. The paper is self-contained and does not fit parameters to data; the algorithms are based on Barvinok's theory, which is appropriate. The claimed fixed-dimension algorithms are valuable if correct. However, the algorithmic theorem and the proof of the main structural theorem contain significant gaps that must be repaired before the results can be considered established.

major comments (3)
  1. [§5, Lemma 5.4] The test set construction in Lemma 5.4 is incomplete and can output the empty set for valid inputs. The construction enumerates only linearly independent subsets of N=(P-P)∩Z^d of size d-1 and takes primitive integer normals to their span. If the lattice points of a full-dimensional rational polytope P lie in a lower-dimensional affine subspace, N has rank r<d-1, no such subset exists, and T=∅, yet aw(P) is finite and attained. For example, in R^3 take P=conv((0,0,0),(2,0,0),(1,1/3,1/3),(1,-1/3,1/3)). The only lattice points in P are (0,0,0),(1,0,0),(2,0,0), so N has rank 1; T is empty. But c=(0,1,0) maps all lattice points to 0, so aw_c(P)=1 and aw(P)=1. Thus Main Theorem 3(b) is unproved as stated. The repair is to enumerate independent subsets of all sizes 1,...,d-1 and take a primitive integer vector orthogonal to each span; the theorem and proof need revision.
  2. [§5, Theorem 5.3] The concluding step of the proof of Theorem 5.3 is incorrect: after computing the univariate generating function h(t) whose monomials correspond to AR_c(P), the proof states that 'evaluating h(t) at t=1 yields precisely the desired count.' In fact h(1) equals the number of lattice points |P∩Z^d|, not the number of distinct values |AR_c(P)|. The support size of a polynomial cannot be read off from its value at 1. Consequently the claimed polynomial-time algorithm for aw_c(P) in Main Theorem 3(a) is not established by the given proof. A different method to count the distinct exponents in the short rational representation of h(t) is needed; alternatively, one may use the Barvinok–Woods projection theorem (Lemma 5.2) and a support-size computation, but this is not supplied.
  3. [§3, Theorem 3.3] The reduction in the proof of Theorem 3.3 is invalid. After setting L=aff(K)∩Z^d and J=K∩L, the proof claims nJ∩Z^d = nK∩Z^d for all n. This is false: if K is a full-dimensional body whose lattice points are collinear, then J is just those finitely many lattice points, and nK∩Z^d contains lattice points that are not n times an original lattice point. For instance, with P as in the counterexample to Lemma 5.4, the point (3,1,1) lies in 3P∩Z^3 but (3,1,1)/3 is not in P∩Z^3, so 3J does not contain it. The argument 'x∈aff(L) so x∈J' confuses the real affine span of L with L itself. This gap undermines the proof of the almost-arithmetic-progression theorem (Main Theorem 1) and the step-size λ used in Main Theorem 2. The statements may be true, but a different reduction (for example, working with the lattice generated by differences of lattice points of all dilates) is required.
minor comments (5)
  1. [§5, Lemma 5.4] 'Barkinov' is a typo for 'Barvinok' in the proof of Lemma 5.4.
  2. [§5, Lemma 5.4] The sentence 'For rational polytopes, |N| can be computed in polynomial time... However, the number of lattice points in P, and thus N, can be exponentially large' should clarify that the cardinality is computable in polynomial time while listing the set may be exponential; as written it reads as a contradiction.
  3. [§4, Theorem 4.8] In the displayed formula for aw(nP), the step size λ should be indexed by the residue class i, since λ depends on the optimal direction c_i^*; the current notation suggests a single λ for all classes.
  4. [§3, Definition 3.1] In the definition of an almost arithmetic progression, the sets A and A' are said to be subsets of [m,m+t]∩(λZ+m) and [M-t',M]∩(λZ+m), respectively, but they should represent the omitted values; the notation could be clarified.
  5. [§2, Proposition 2.1(d)] The phrase 'giving us our final claim' ends the proof of the maximal arithmetic width, but the sentence preceding it is missing a verb; please revise for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main results are derived from the paper's own definitions and external Barvinok machinery; the sole self-citation is a non-load-bearing remark.

full rationale

The paper's central claims are self-contained rather than circular. Main Theorem 1 follows from Theorem 3.3, whose proof constructs the constants t, t', and lambda from the geometry of K and the lattice, using Bezout's identity to guarantee lattice points in affine hyperplanes; no parameter is fitted from the quantity being predicted. Main Theorem 2 derives eventual quasilinearity from Lemma 4.1 and the recurrence in Theorem 4.2, and the proof identifies the linear coefficient D(M-m)/lambda without assuming the conclusion. Main Theorem 3 uses Barvinok's external short-rational-generating-function theory (Lemmas 5.1 and 5.2) to count distinct monomials, and Lemma 5.4 constructs a test set from the difference set N=(P-P) cap Z^d; the test directions are derived, not fitted to the output. Main Theorem 4 is proven by explicit examples. The only self-citation is Remark 4.5, where the authors note that applying Lemma 4.1 to numerical semigroup polytopes gives an alternative proof of [28, Theorem 4.2]; this citation is not used to prove any new claim in the present paper, so it is not load-bearing and does not create circularity. The skeptical concern about Lemma 5.4 is a correctness gap: for full-dimensional rational polytopes whose lattice points are collinear, the enumerated independent subsets of size d-1 do not exist, so T can be empty even though a minimizing direction exists. That is a mathematical flaw in the theorem's proof, not a case of the paper's derivation reducing to its own inputs, and under the instructions it is recorded as correctness risk rather than circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central results are derivations with no fitted parameters or invented entities. The robustness of the algorithmic claims is not a matter of hidden assumptions; the issue is that the proofs contain gaps and errors.

assumptions (4)
  • standard math Covering radius of a lattice in a rational subspace is finite, used to choose r in Theorem 3.3 such that every translate of B_r(0)∩H contains a lattice point.
    The proof of Theorem 3.3 relies on the existence of a uniform r so that every hyperplane slice of the affine hull contains a lattice point.
  • standard math Bezout's identity for primitive integer vectors, so for primitive c and any integer j there exists x in Z^d with c^T x = j.
    Used in Theorem 3.3 to ensure that every value in the arithmetic progression has a corresponding hyperplane with integer points.
  • standard math Barvinok's theorem on short rational generating functions and the Barvinok-Woods projection theorem.
    The algorithms in Section 5 assume these fixed-dimension polynomial-time tools for encoding and manipulating lattice point sets.
  • domain assumption The domain assumption that P is a rational polytope with denominator D and that K's affine span contains a lattice point.
    Rationality ensures D p_m and D p_M are integer vectors in Lemma 4.1; the lattice point condition is handled by integer translation in Theorem 3.3.

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Pith. "Pith review of An arithmetic measure of width for convex bodies." pith.science (2026). https://pith.science/paper/RPXJZWJN

@misc{pith2026250904726,
  author       = {Pith},
  title        = {Pith review of: An arithmetic measure of width for convex bodies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RPXJZWJN}},
  note         = {Machine review of arXiv:2509.04726}
}
read the original abstract

We introduce the arithmetic width of a convex body, defined as the number of distinct values a linear functional attains on the lattice points within the body. Arithmetic width refines lattice width by detecting gaps in the lattice point distribution and always provides a natural lower bound. We show that for large dilates of a convex body, the attained values form an arithmetic progression with only a bounded number of omissions near the extremes. For rational polytopes, we show that the arithmetic width grows eventually quasilinearly in the dilation parameter, with optimal directions reoccurring periodically. Lastly, we present algorithms to compute the arithmetic width. These results build new connections with discrete geometry, integer programming, and additive combinatorics.

Figures

Figures reproduced from arXiv: 2509.04726 by the authors.

Figure 1
Figure 1. A convex body K depicted with a hyperplane contributing to lattice width but containing no lattice points in K (left), and the polytope P in Example 1.3 (right). those gaps. In contrast, the arithmetic width is a finer invariant than the integer or Euclidean widths as it accounts for these missing lattice points in our polytope in fixed directions. Definition 1.1. Fix a nonzero c ∈ Z d . The hyperplane cover of K ∩ … view at source ↗
Figure 2
Figure 2. The constants n, t, and t ′ from Main Theorem 1. 1.1. Main results. The main results of this paper comprise of four main theorems. The first theorem is a structure theorem for the arithmetic range: for sufficiently large dilates of a convex body, the arithmetic range has a highly regular structure, forming an “almost arithmetic progression” - an arithmetic progression except for finitely many exceptions near the ext… view at source ↗
Figure 3
Figure 3. Illustration of the proof of Theorem 3.3. 4. The arithmetic width of rational polytopes In this section, we will build upon our results from Theorem 3.3 for our analysis of the arithmetic width of dilates of convex polytopes. We begin this section with definitions that explore the complement of the arithmetic range: the sets of gaps. From there, we compare gaps across subsequent dilates of polytopes, which gives eno… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Dilations of the polytope P from Example 4.7, partitioned based on their optimal directions. Proof. Without loss of generality, assume n1 > n2 such that n1, n2 ≫ 0, and fix k ∈ Z>0 such that n1 = n2 + kD. Then, applying the recurrence relationship in Equation (1), we s…

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