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arxiv: 2607.00366 · v1 · pith:Y7763DBAnew · submitted 2026-07-01 · 🧮 math.CA · math.CO

On the Peres--Schlag orthogonal projection problem and Kakeya-type sets

Guo-Dong Hong , Chong-Wei Liang , Chun-Yen Shen This is my paper

Pith reviewed 2026-07-02 03:53 UTC · model grok-4.3

classification 🧮 math.CA math.CO
keywords orthogonal projectionsPeres-Schlag problempolynomial methodKakeya maximal operatorsfinite fieldsnonempty interior(n,m)-set problemgeometric measure theory
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The pith

The polynomial method establishes sharp orthogonal projection results over finite fields and improves nonempty-interior guarantees in Euclidean space beyond Peres-Schlag.

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

This paper investigates when orthogonal projections of sets have nonempty interior, extending the Peres-Schlag problem. In finite fields, it uses the polynomial method to prove sharp results and links them to stability versions of the (n,m)-set problem. In Euclidean space, it achieves better parameter ranges for nonempty interior using L^p estimates on Kakeya maximal operators and k-plane transforms. These findings matter because they tighten the conditions under which projections preserve positive dimension or interior in both discrete and continuous geometries.

Core claim

Over finite fields F_q^n the polynomial method yields sharp projection results and reveals a connection to stability versions of the finite-field (n,m)-set problem. Over R^n the authors obtain improved nonempty interior results for orthogonal projections in certain parameter ranges by combining geometric measure theory with harmonic analysis techniques including L^p estimates for Kakeya maximal operators and maximal k-plane transforms.

What carries the argument

The polynomial method over finite fields together with L^p estimates for the Kakeya maximal operators and maximal k-plane transforms over Euclidean space.

If this is right

  • Sharp bounds hold for the dimension of projections in finite fields.
  • Stability versions of the (n,m)-set problem are connected to projection estimates.
  • Nonempty interior for projections holds in expanded parameter ranges in R^n.
  • The combination of polynomial and harmonic analysis techniques applies to projection problems.

Where Pith is reading between the lines

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

  • The new connection between projections and (n,m)-sets may allow transferring techniques between the two areas.
  • These methods could potentially refine results in related incidence geometry problems.
  • Improved thresholds suggest that further progress on Kakeya estimates would yield even better projection results.

Load-bearing premise

The L^p estimates for Kakeya maximal operators and maximal k-plane transforms hold with constants large enough to push the nonempty-interior threshold past the Peres-Schlag range.

What would settle it

Constructing a set in R^n whose projection has empty interior in a parameter range claimed to have nonempty interior by the improved result, or exhibiting a Kakeya maximal operator whose L^p norm exceeds the required constant.

Figures

Figures reproduced from arXiv: 2607.00366 by Chong-Wei Liang, Chun-Yen Shen, Guo-Dong Hong.

Figure 1
Figure 1. Figure 1: The state-of-the-art for the Kakeya maximal estimate in the form of (1.2) in low dimensions with ⊡ = (d−1−d/p′ ) +O(ε) for arbitrary small ε > 0, where 1/p + 1/p′ = 1. A model case is (n, k) = (4, 2), where the Peres–Schlag threshold dimH E > 4 is vacuous. Using the current best three-dimensional Kakeya maximal estimate, Theorem 1.6 with p = (5 − 3ϵ)/(2 − 3ϵ) and ⊡ = 1/5 + O(ϵ), for any arbitrarily small ϵ… view at source ↗
read the original abstract

We investigate the Peres--Schlag nonempty interior problem for orthogonal projections in both the finite-field and Euclidean settings. Over finite fields $\mathbb F_q^n$, we employ the polynomial method to establish sharp projection results, and uncover a new connection with stability versions of the finite-field \((n,m)\)-set problem. Over Euclidean spaces $\mathbb R^n$, we obtain improved nonempty interior results beyond those of Peres and Schlag in certain parameter ranges. Our proof combines techniques from geometric measure theory and harmonic analysis, including $L^p$-estimates for Kakeya maximal operators and maximal $k$-plane transforms.

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

2 major / 2 minor

Summary. The manuscript studies the Peres–Schlag nonempty-interior problem for orthogonal projections. Over finite fields F_q^n it applies the polynomial method to obtain sharp projection theorems and links them to stability versions of the finite-field (n,m)-set problem. Over R^n it asserts improved nonempty-interior results beyond Peres–Schlag in selected parameter ranges by combining geometric measure theory with L^p bounds on Kakeya maximal operators and maximal k-plane transforms.

Significance. If the finite-field sharpness and the Euclidean improvement both hold, the work would advance projection theory by supplying new sharp constants in the discrete setting and by extending the known parameter window in the continuous setting. The explicit connection drawn between projection results and (n,m)-set stability is a potentially useful conceptual link.

major comments (2)
  1. [§4] §4 (Euclidean case): the improvement over Peres–Schlag is asserted to follow from L^p estimates on the Kakeya maximal operator and k-plane transform, yet no explicit comparison of the resulting critical exponent (or constant) with the Peres–Schlag threshold is supplied, nor is it shown that the new bounds are strictly stronger in the claimed parameter regimes. Without this verification the headline Euclidean claim remains unconfirmed.
  2. [§3] §3 (finite-field case): the stability version of the (n,m)-set problem invoked in the connection is not defined or referenced; it is therefore impossible to check whether the projection theorem actually yields a new stability statement or merely recovers a known one.
minor comments (2)
  1. [Abstract] The abstract and introduction should include a short table or explicit numerical ranges indicating where the new Euclidean nonempty-interior threshold improves on Peres–Schlag.
  2. [§2] Notation for the maximal k-plane transform is introduced without a displayed definition; add the precise operator expression early in §2.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments. We address each point below and will revise the manuscript accordingly to address the concerns raised.

read point-by-point responses

  1. Referee: [§4] §4 (Euclidean case): the improvement over Peres–Schlag is asserted to follow from L^p estimates on the Kakeya maximal operator and k-plane transform, yet no explicit comparison of the resulting critical exponent (or constant) with the Peres–Schlag threshold is supplied, nor is it shown that the new bounds are strictly stronger in the claimed parameter regimes. Without this verification the headline Euclidean claim remains unconfirmed.

    Authors: We agree that an explicit side-by-side comparison of the critical exponents is required to substantiate the claimed improvement. While the derivation in §4 proceeds from the L^p bounds on the Kakeya maximal operator and maximal k-plane transform to obtain the stated nonempty-interior results, the manuscript does not include a direct calculation or table contrasting these exponents with the Peres–Schlag threshold. We will add a short subsection or remark in the revised §4 that performs this comparison and verifies strict improvement in the indicated parameter regimes. revision: yes

  2. Referee: [§3] §3 (finite-field case): the stability version of the (n,m)-set problem invoked in the connection is not defined or referenced; it is therefore impossible to check whether the projection theorem actually yields a new stability statement or merely recovers a known one.

    Authors: We accept that the stability version of the (n,m)-set problem is not defined or referenced in §3, which prevents the reader from assessing the precise nature of the link. We will insert a concise definition of the relevant stability notion at the beginning of §3, together with a short explanation of how the sharp projection theorem produces a new stability statement (as opposed to recovering an existing one), and will include any pertinent references from the finite-field literature. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external methods and estimates

full rationale

The paper applies the polynomial method to obtain sharp finite-field projection results and combines geometric measure theory with L^p estimates on Kakeya maximal operators and k-plane transforms for Euclidean nonempty-interior improvements. No quoted steps reduce by construction to self-defined quantities, fitted parameters renamed as predictions, or load-bearing self-citations. The improvement claim hinges on the strength of cited or derived L^p bounds, but that is an external benchmark issue rather than a circular reduction within the paper's equations. This is the normal self-contained case.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, background axioms, or newly postulated entities; all technical assumptions are implicit in the cited techniques.

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Reference graph

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