On the number of directions formed by Cartesian products in $\mathbb{F}_{p^2}^2$

Ali Mohammadi

arxiv: 2605.15727 · v1 · pith:JL6XW5JUnew · submitted 2026-05-15 · 🧮 math.CO · math.NT

On the number of directions formed by Cartesian products in mathbb{F}_(p²)²

Ali Mohammadi This is my paper

Pith reviewed 2026-05-20 17:41 UTC · model grok-4.3

classification 🧮 math.CO math.NT

keywords directionsCartesian productsfinite fieldsaffine planescombinatoricssubfield structuremultiplicity arguments

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The pith

Cartesian products A×A in the plane over F_{p squared} determine a large number of directions for sets A sized between p to the 2/3 and p, provided they avoid affine copies of the subfield F_p.

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

The paper proves a lower bound for the number of distinct directions determined by points in the Cartesian product A times A inside the affine plane over the finite field with p squared elements. The bound applies specifically when the size of A exceeds p to the power of two thirds but stays below p, and when A is not contained in any affine line that comes from the subfield with p elements. The argument uses a description of how the directions must be arranged unless they reflect a subfield structure, in which case a particular direction is selected to make the counting of multiplicities work out explicitly. This helps clarify the minimal number of directions such product sets can produce in these finite geometries.

Core claim

The author establishes that unless the set of directions determined by A×A has closure properties that force A to have subfield structure, there exists a direction for which the algebraic multiplicity parameter allows a concrete lower bound on the total number of directions via a known counting theorem.

What carries the argument

A structural property of the direction set from the Cartesian product that either forces subfield containment or permits an explicit multiplicity in the direction counting argument.

If this is right

The number of directions from A×A is bounded from below in terms of p and |A| for the given size range.
Product sets avoid having too few directions outside of subfield configurations.
The result extends previous work by handling the intermediate size range through this conditional multiplicity choice.

Where Pith is reading between the lines

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

If the lower bound is close to optimal, it would suggest that subfield copies achieve near-minimal directions in this setting.
This technique of selecting a direction to make parameters explicit could apply to similar problems in other finite fields or geometries.
Future work might try to remove the size restrictions or find matching upper bounds through explicit constructions.

Load-bearing premise

The structural result on the set of directions from Cartesian products applies in a form that does not add restrictions preventing the use of the multiplicity-based lower bound.

What would settle it

A counterexample would be a set A of size roughly p to the three quarters power, not contained in any affine copy of F_p, such that A times A determines fewer directions than the lower bound claims.

read the original abstract

We prove a lower bound on the number of directions determined by Cartesian products $A\times A$ in the affine plane over the finite field $\mathbb F_{p^2}$. Our lower bound holds for sets of size $p^{2/3}<|A|<p$, which are not contained in any affine copy of $\mathbb F_p$. The proof combines a structural result of Li and Roche-Newton on the set of directions formed by Cartesian products with a lower bound of Fancsali, Sziklai and Tak\'{a}ts. A key step shows that, unless the set of directions exhibits closure properties forcing subfield structure, one obtains a direction for which an algebraic multiplicity parameter in the latter theorem can be made explicit.

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 refines the direction lower bound for A×A in F_{p²} by making the multiplicity parameter explicit once subfield closure is ruled out.

read the letter

The paper's main takeaway is a lower bound for the number of directions from Cartesian products A×A over F_{p²} in the range p^{2/3} < |A| < p, assuming A avoids affine subfield copies. They achieve this by linking Li and Roche-Newton's structural result on directions to the Fancsali-Sziklai-Takáts bound, with the key step being an explicit algebraic multiplicity for a suitable direction when subfield closure does not occur. This combination is handled directly and the non-subfield assumption is used to avoid the degenerate case. The paper stays close to the two cited theorems and adds the observation that makes the multiplicity usable. The potential soft spot is whether the direction from the structural theorem satisfies all the conditions required by the multiplicity result without extra verification. The abstract indicates they handle closure properties to rule out subfields, but the precise transfer of the multiplicity parameter needs checking in the full argument. Readers working on finite field incidence geometry or direction problems will get value from this. It is a targeted refinement rather than a broad new method. I would send it to peer review. The technical step is specific enough that referees can assess whether the bound follows cleanly.

Referee Report

1 major / 1 minor

Summary. The manuscript proves a lower bound on the number of directions determined by Cartesian products A×A in the affine plane over the finite field F_{p²} for sets A with p^{2/3} < |A| < p that are not contained in any affine copy of F_p. The proof combines a structural result of Li and Roche-Newton on the set of directions formed by Cartesian products with a lower bound of Fancsali, Sziklai and Takáts, via a key step showing that unless the direction set exhibits closure properties forcing subfield structure, one obtains a direction for which an algebraic multiplicity parameter can be made explicit.

Significance. If the central combination of the cited structural theorem with the multiplicity bound holds without additional restrictions in the quadratic extension, the result supplies a non-trivial lower bound in an intermediate size regime for direction sets of Cartesian products. The approach of extracting an explicit multiplicity parameter from the Li-Roche-Newton structure is a clear strength when the non-degeneracy conditions align.

major comments (1)

Abstract and key step: The claim that the direction d obtained from the Li-Roche-Newton structural result automatically satisfies the hypotheses of the Fancsali-Sziklai-Takáts multiplicity theorem (allowing the algebraic multiplicity parameter to be made explicit) requires explicit verification. In particular, confirm that the algebraic relation guaranteed in F_{p²} implies the precise non-vanishing or degree conditions needed for the multiplicity bound, without unstated non-degeneracy assumptions that might fail in the quadratic extension.

minor comments (1)

The abstract refers to 'an algebraic multiplicity parameter in the latter theorem'; adding a brief inline definition or reference to the exact parameter (e.g., the multiplicity m(d)) when first introduced would improve readability for readers unfamiliar with the Fancsali-Sziklai-Takáts result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the insightful comment. We address the major comment in detail below.

read point-by-point responses

Referee: Abstract and key step: The claim that the direction d obtained from the Li-Roche-Newton structural result automatically satisfies the hypotheses of the Fancsali-Sziklai-Takáts multiplicity theorem (allowing the algebraic multiplicity parameter to be made explicit) requires explicit verification. In particular, confirm that the algebraic relation guaranteed in F_{p²} implies the precise non-vanishing or degree conditions needed for the multiplicity bound, without unstated non-degeneracy assumptions that might fail in the quadratic extension.

Authors: We appreciate the referee drawing attention to this point. The direction d extracted via the Li-Roche-Newton structural theorem is chosen precisely so that the associated algebraic relation (a polynomial equation over F_{p²}) has degree and non-vanishing properties compatible with the Fancsali-Sziklai-Takáts multiplicity bound. Because A is assumed not to lie in any affine copy of F_p, the relation cannot degenerate to a subfield equation; this forces the multiplicity parameter to be positive and explicit without additional assumptions. We will add a short paragraph in the revised manuscript (near the key step combining the two cited results) that verifies the transfer of these non-degeneracy conditions to the quadratic extension. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies independent external theorems

full rationale

The paper's central argument invokes a structural result on directions from Li and Roche-Newton together with a multiplicity lower bound from Fancsali, Sziklai and Takáts. These are distinct external citations whose hypotheses are checked against the paper's assumptions on A (size range and non-containment in affine F_p). The bridging step that produces an explicit algebraic multiplicity parameter follows directly from the cited structural dichotomy and does not reduce to a fit, redefinition, or self-citation chain within the present work. No equations or claims in the abstract or key-step description exhibit definitional equivalence or statistical forcing. The derivation therefore remains self-contained against the cited benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; the paper invokes two external theorems whose assumptions are treated as given. No free parameters or new entities are visible in the summary.

axioms (2)

domain assumption Structural theorem of Li and Roche-Newton on directions determined by Cartesian products holds in F_{p²}²
Cited as the first main ingredient; invoked to classify direction sets that do or do not force subfield structure.
domain assumption Lower bound of Fancsali, Sziklai and Takáts on directions with algebraic multiplicity parameter applies once a suitable direction is identified
Cited as the second main ingredient; the paper's key step is to make the multiplicity parameter explicit for at least one direction.

pith-pipeline@v0.9.0 · 5651 in / 1375 out tokens · 90339 ms · 2026-05-20T17:41:48.620528+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

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S. Ball, The number of directions determined by a function over a finite field,J. Combin. Theory Ser. A 104(2003), no. 2, 341–350

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Di Benedetto, J

D. Di Benedetto, J. Solymosi, and E. P. White, On the directions determined by a Cartesian product in an affine Galois plane,Combinatorica41(2021), no. 6, 755–763

work page 2021
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S. L. Fancsali, P. Sziklai, and M. Tak´ ats, The number of directions determined by less thanqpoints,J. Algebraic Combin.37(2013), 27–37

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Li and O

L. Li and O. Roche-Newton, An improved sum-product estimate for general finite fields,SIAM J. Discrete Math.25(2011), no. 3, 1285–1296

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N. H. Katz and C.-Y. Shen, A slight improvement to Garaev’s sum-product estimate,Proc. Amer. Math. Soc.136(2008), no. 7, 2499–2504

work page 2008
[6]

Macourt, G

S. Macourt, G. Petridis, I. D. Shkredov, and I. E. Shparlinski, Bounds of trilinear and trinomial expo- nential sums,SIAM J. Discrete Math.34(2020), no. 4, 2124–2136

work page 2020
[7]

Petridis and I

G. Petridis and I. E. Shparlinski, Bounds of trilinear and quadrilinear exponential sums,J. Aust. Math. Soc.138(2019), 613–641

work page 2019
[8]

R´ edei,L¨ uckenhafte Polynome ¨ uber endlichen K¨ orpern, Birkh¨ auser-Verlag, Basel, 1970

L. R´ edei,L¨ uckenhafte Polynome ¨ uber endlichen K¨ orpern, Birkh¨ auser-Verlag, Basel, 1970. (English trans- lation:Lacunary Polynomials over Finite Fields, North-Holland, Amsterdam, 1973)

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Tao and V

T. Tao and V. Vu,Additive Combinatorics, Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge University Press, Cambridge, 2006. Email address:ali.mohammadi.np@gmail.com

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[1] [1]

Ball, The number of directions determined by a function over a finite field,J

S. Ball, The number of directions determined by a function over a finite field,J. Combin. Theory Ser. A 104(2003), no. 2, 341–350

work page 2003

[2] [2]

Di Benedetto, J

D. Di Benedetto, J. Solymosi, and E. P. White, On the directions determined by a Cartesian product in an affine Galois plane,Combinatorica41(2021), no. 6, 755–763

work page 2021

[3] [3]

S. L. Fancsali, P. Sziklai, and M. Tak´ ats, The number of directions determined by less thanqpoints,J. Algebraic Combin.37(2013), 27–37

work page 2013

[4] [4]

Li and O

L. Li and O. Roche-Newton, An improved sum-product estimate for general finite fields,SIAM J. Discrete Math.25(2011), no. 3, 1285–1296

work page 2011

[5] [5]

N. H. Katz and C.-Y. Shen, A slight improvement to Garaev’s sum-product estimate,Proc. Amer. Math. Soc.136(2008), no. 7, 2499–2504

work page 2008

[6] [6]

Macourt, G

S. Macourt, G. Petridis, I. D. Shkredov, and I. E. Shparlinski, Bounds of trilinear and trinomial expo- nential sums,SIAM J. Discrete Math.34(2020), no. 4, 2124–2136

work page 2020

[7] [7]

Petridis and I

G. Petridis and I. E. Shparlinski, Bounds of trilinear and quadrilinear exponential sums,J. Aust. Math. Soc.138(2019), 613–641

work page 2019

[8] [8]

R´ edei,L¨ uckenhafte Polynome ¨ uber endlichen K¨ orpern, Birkh¨ auser-Verlag, Basel, 1970

L. R´ edei,L¨ uckenhafte Polynome ¨ uber endlichen K¨ orpern, Birkh¨ auser-Verlag, Basel, 1970. (English trans- lation:Lacunary Polynomials over Finite Fields, North-Holland, Amsterdam, 1973)

work page 1970

[9] [9]

Sz˝ onyi, Around R´ edei’s theorem,Discrete Math.208/209(1999), 557–575

T. Sz˝ onyi, Around R´ edei’s theorem,Discrete Math.208/209(1999), 557–575

work page 1999

[10] [10]

Tao and V

T. Tao and V. Vu,Additive Combinatorics, Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge University Press, Cambridge, 2006. Email address:ali.mohammadi.np@gmail.com

work page 2006