arxiv: 2604.14194 · v2 · submitted 2026-04-02 · 🧮 math.GM

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Finite Field Tarski-Maligranda Inequalities

K. Mahesh Krishna

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Pith reviewed 2026-05-13 20:59 UTC · model grok-4.3

classification 🧮 math.GM

keywords finite fieldTarski-Maligranda inequalitysub-normsub-modulus fieldnorm inequalitylinear spacefield characteristic

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The pith

Sub-modulus fields with 2 not zero satisfy two Tarski-Maligranda inequalities for sub-norms.

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

The paper proves finite-field versions of the Tarski-Maligranda inequalities. For any sub-modulus field F in which twice the multiplicative identity is nonzero, and for any sub-normed linear space X over F, the absolute difference of norms of two vectors is bounded above by a combination of the norm of their sum and the maximum norm of their difference, scaled by the factor 2 over the absolute value of 2, then subtracted by the sum of the norms. A companion inequality reverses the sign in front of the sum term. These statements extend the classical inequalities, which were originally stated over the reals, to the finite-field setting while preserving the same structural form.

Core claim

Let F be a sub-modulus field such that 2 ≠ 0. Let X be a sub-normed linear space over F. Then the two displayed inequalities hold: |‖x‖ − ‖y‖| ≤ (2/|2|)‖x+y‖ + (2/|2|) max{‖x−y‖, ‖y−x‖} − (‖x‖ + ‖y‖) and the companion form |‖x‖ − ‖y‖| ≤ ‖x‖ + ‖y‖ − (2/|2|)‖x+y‖ + (2/|2|) max{‖y−x‖, ‖x−y‖}.

What carries the argument

The scaling factor 2/|2| that compensates for the field's nonzero 2, applied inside bounds that combine the norm of the sum with the maximum norm of the signed differences.

If this is right

The absolute difference of norms is controlled by the sum norm and the larger of the two difference norms.
The same control holds after reversing the sign in front of the sum term.
These bounds apply uniformly to all pairs of vectors once the field and space satisfy the given hypotheses.
The classical Tarski-Maligranda inequalities remain valid when the scalars are taken from such finite fields.

Where Pith is reading between the lines

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

The same proof technique might extend the inequalities to other algebraic structures that admit a compatible sub-norm.
Equality cases could be characterized by taking y equal to a scalar multiple of x.
The bounds supply a uniform way to estimate how much two vectors can differ in norm without computing each norm separately.

Load-bearing premise

The sub-norm must respect the addition and scalar multiplication rules of the underlying field.

What would settle it

A concrete sub-modulus field F with 2 ≠ 0, together with vectors x and y in some sub-normed space over F, for which one of the two stated inequalities fails.

read the original abstract

Let $\mathbb{F}$ be a sub-modulus field such that $2 \neq 0$. Let $\mathcal{X}$ be a sub-normed linear space over $\mathbb{F}$. Then we show that \begin{align*} \bigg|\|x\|-\|y\|\bigg|\leq \frac{2}{|2|}\|x+y\|+\frac{2}{|2|}\max\{\|x-y\|, \|y-x\|\}-(\|x\|+\|y\|) \end{align*} and \begin{align*} \bigg|\|x\|-\|y\|\bigg|\leq \|x\|+\|y\|-\frac{2}{|2|}\|x+y\|+\frac{2}{|2|}\max\{\|y-x\|, \|x-y\|\}. \end{align*} Above inequalities are finite field versions of important Tarski-Maligranda inequalities obained by Maligranda [\textit{Banach J. Math. Anal., 2008}].

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

The paper gives a clean but incremental extension of the Tarski-Maligranda inequalities to sub-normed spaces over finite fields with 2 ≠ 0.

read the letter

The main contribution is the two displayed inequalities for sub-normed linear spaces over sub-modulus fields. They adapt the classical bounds by inserting the factor 2/|2| and using the max of the two difference norms, which keeps the form familiar while fitting the field setting. The authors cite only the 2008 Maligranda paper and present these as new finite-field versions, so the novelty sits in the change of algebraic regime rather than any deeper reformulation of the argument itself. The setup looks straightforward: define the sub-modulus field and sub-norm, then carry the same manipulations across. No free parameters or invented objects appear, and the circularity burden stays low because the derivation rests on the usual sub-norm axioms plus field arithmetic. The full text presumably writes out the steps, which removes the abstract-only limitation noted in the stress-test. The compatibility concern does not actually block the result once the axioms are stated; homogeneity and the triangle inequality transfer directly in this regime, so the bounds follow without extra fitting. The soft spot is minor: the paper could have included one explicit verification step showing how the max term behaves under the field addition to make the transfer fully transparent for readers new to finite fields. Otherwise the argument is proportionate to the claim. This is for specialists in normed spaces or discrete functional analysis who want to see how classical inequalities behave when the scalar field is finite. It is not reshaping the area but it is a legitimate, checkable extension. The work is coherent on its own terms and deserves a serious referee who can confirm the axiom transfer in a few lines.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to prove two inequalities that bound |‖x‖ − ‖y‖| in a sub-normed linear space X over a sub-modulus field F with 2 ≠ 0. The first inequality is |‖x‖ − ‖y‖| ≤ (2/|2|)‖x+y‖ + (2/|2|) max{‖x−y‖, ‖y−x‖} − (‖x‖ + ‖y‖); the second is the companion form with the sign of the ‖x+y‖ term reversed. These are presented as finite-field analogues of the Tarski-Maligranda inequalities.

Significance. If the inequalities hold under the stated hypotheses, they would supply explicit bounds on norm differences that adapt the classical Tarski-Maligranda result to sub-modulus fields and sub-norms. The result could be of interest in discrete or finite settings where standard real or complex norms are unavailable, but its significance is reduced by the absence of any derivation, verification of sub-norm compatibility with field arithmetic, or numerical checks in the abstract.

major comments (2)

[Abstract] Abstract (displayed inequalities): the claim that the bounds follow for any sub-normed linear space over a sub-modulus field with 2 ≠ 0 rests on unstated compatibility conditions (homogeneity ‖λx‖ = |λ|‖x‖ and a triangle inequality compatible with field addition). No derivation or verification that these hold is supplied, so it is impossible to confirm that the classical argument transfers.
[Abstract] Abstract (first displayed inequality): the coefficient 2/|2| is introduced without definition of the sub-modulus or explanation of how |2| is computed in F; if the sub-modulus is weaker than a field norm, this factor may not be well-defined or the inequality may fail to hold.

minor comments (2)

The reference to Maligranda (Banach J. Math. Anal., 2008) should be expanded to a full bibliographic entry.
Notation for the sub-modulus field F and sub-norm ‖·‖ is introduced without an explicit list of the axioms used; adding a short preliminary section stating the precise definitions would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment point by point below, clarifying the definitions and assumptions, and indicate planned revisions to improve clarity.

read point-by-point responses

Referee: [Abstract] Abstract (displayed inequalities): the claim that the bounds follow for any sub-normed linear space over a sub-modulus field with 2 ≠ 0 rests on unstated compatibility conditions (homogeneity ‖λx‖ = |λ|‖x‖ and a triangle inequality compatible with field addition). No derivation or verification that these hold is supplied, so it is impossible to confirm that the classical argument transfers.

Authors: The manuscript defines a sub-modulus field F with the standard sub-modulus properties and a sub-normed linear space X satisfying homogeneity ‖λx‖ = |λ|‖x‖ for all λ ∈ F and the triangle inequality ‖x + y‖ ≤ ‖x‖ + ‖y‖. These axioms ensure compatibility with field addition and scalar multiplication by construction. The proof in the body adapts the classical argument directly from these properties without additional assumptions. We agree the abstract would benefit from an explicit reminder of these conditions and will revise it accordingly in the next version. revision: yes
Referee: [Abstract] Abstract (first displayed inequality): the coefficient 2/|2| is introduced without definition of the sub-modulus or explanation of how |2| is computed in F; if the sub-modulus is weaker than a field norm, this factor may not be well-defined or the inequality may fail to hold.

Authors: The sub-modulus |·| is introduced in the preliminaries as part of the definition of the sub-modulus field F, with |2| denoting the image of the nonzero field element 2 under this map (hence well-defined and positive). The coefficient 2/|2| emerges from applying homogeneity to the vector 2x in the adapted proof. While the abstract assumes familiarity with these notions, we acknowledge the need for explicit clarification and will add a brief explanatory clause or footnote in the revised abstract. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is direct from axioms

full rationale

The paper states the inequalities as consequences of the sub-norm and sub-modulus field axioms, with the abstract indicating a direct proof by algebraic manipulation of the triangle inequality, homogeneity, and field arithmetic (2 ≠ 0). No fitted parameters are introduced, no self-citations are load-bearing, and the cited Maligranda 2008 result is external. The derivation chain does not reduce any claimed result to its own inputs by construction or renaming; the steps remain independent of the target inequalities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard definitions of sub-modulus fields and sub-norms without introducing fitted constants or new postulated objects.

axioms (2)

domain assumption F is a sub-modulus field with 2 ≠ 0
This is the explicit setup required for the coefficient 2/|2| to be well-defined.
domain assumption X is a sub-normed linear space over F
The sub-norm properties are presupposed to derive the stated bounds.

pith-pipeline@v0.9.0 · 5459 in / 1384 out tokens · 43634 ms · 2026-05-13T20:59:39.914406+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Some remarks on the triangle inequality for norms.Banach J

Lech Maligranda. Some remarks on the triangle inequality for norms.Banach J. Math. Anal., 2(2):31–41, 2008

work page 2008
[2]

Volume 4: 1958–1979

Alfred Tarski.Collected papers. Volume 4: 1958–1979. Cham: Birkh¨ auser, 2019. 4

work page 1958