Peres--Schlag's nonempty-interior problem and a shifted-product variant for product sets

Chong-Wei Liang; Chun-Yen Shen; Guo-Dong Hong

arxiv: 2607.00372 · v1 · pith:EPE334QAnew · submitted 2026-07-01 · 🧮 math.CO · math.CA

Peres--Schlag's nonempty-interior problem and a shifted-product variant for product sets

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

Pith reviewed 2026-07-02 11:16 UTC · model grok-4.3

classification 🧮 math.CO math.CA

keywords finite fieldssumsetsproduct setscovering problemsHausdorff dimensionadditive combinatoricsPeres-Schlag problem

0 comments

The pith

In finite fields, sets A larger than p to the 3/(2n-1) power have n scaled copies summing to the whole field.

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

The paper proves that for n at least 3 and any small η, any subset A of the finite field F_p with size at least p to the power 3/(2n-1) plus η admits coefficients t1 through tn such that the sum of the scaled copies ti A equals every element of F_p. This beats the exponent 2/n that follows from treating the n-fold product set directly. The identical size threshold suffices when the sums are replaced by the product of n shifted copies (t1 plus A) through (tn plus A). Over the reals the same dimension threshold 2/n guarantees that some shifted product of a Borel set A contains a nonempty open interval.

Core claim

For n≥3 and every η>0, whenever |A| ≫_{n,η} p^{3/(2n-1)+η}, there exist t1,...,tn in F_p such that t1A + ⋯ + tnA = F_p. The same density scale holds for the shifted-product covering (t1+A)⋯(tn+A). In R, if A is Borel with dim_H A > 2/n then some shifted product contains a nonempty open interval.

What carries the argument

The finite-field covering argument achieving the exponent 3/(2n-1) for the existence of coefficients making a one-dimensional linear image of A^n surjective onto F_p.

If this is right

The same density threshold works for the shifted-product version of the covering problem.
A separate two-dimensional near-half-density covering result holds in the finite-field setting.
The Euclidean statement follows for Borel sets whose Hausdorff dimension exceeds 2/n.

Where Pith is reading between the lines

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

The improvement may extend to other families of linear forms or to higher-dimensional analogues of the covering problem.
The shared threshold for sums and products hints at deeper sum-product type interactions that could be tested in small fields.

Load-bearing premise

Some additive-combinatorial or Fourier-analytic estimate improves the covering exponent from the naive product bound 2/n to 3/(2n-1).

What would settle it

An explicit set A in F_p whose cardinality is slightly larger than p to the 3/(2n-1) but for which no choice of the n coefficients makes the weighted sum or shifted product equal the whole field.

read the original abstract

We study finite-field analogues of the Peres--Schlag nonempty-interior problem for product sets. Given \(A\subseteq\mathbb F_p\), we ask when a suitable one-dimensional linear image of \(A^n\) is full; equivalently, when there exist coefficients \(t_1,\ldots,t_n\in\mathbb F_p\) such that \[ t_1A+\cdots+t_nA=\mathbb F_p. \] For \(n\ge3\), we prove that, for every \(\eta>0\), this holds whenever \[ |A|\gg_{n,\eta} p^{\frac{3}{2n-1}+\eta}. \] This improves the exponent predicted by the direct product-set analogue of the Peres--Schlag threshold, namely \(|A|\gg p^{2/n}\). We also prove a two-dimensional near-half-density result. Motivated by sum-product phenomena, we also introduce and study a product-type variant in which linear forms are replaced by shifted product maps. We prove finite-field covering results for shifted products \[ (t_1 + A)(t_2 + A)\cdots(t_n + A) \] at the same density scale as in the linear case. Finally, we prove a Euclidean shifted-product analogue: if \(A\subseteq\mathbb R\) is Borel and \(\dim_H A>2/n\), then some shifted product of \(n\) copies of \(A\) contains a nonempty open interval.

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

They improve the Peres-Schlag covering threshold in F_p from the naive 2/n to 3/(2n-1) via iterated bilinear estimates, and match the same scale for the shifted-product version.

read the letter

The core advance is a genuine drop in the exponent for when n linear images of A cover F_p, from the direct-product prediction of 2/n down to 3/(2n-1) plus any eta, for n at least 3. The same density works for the shifted-product covering (t1 + A) ... (tn + A). They also give a Euclidean statement that dim_H A > 2/n forces some shifted product to contain an interval.

The argument is laid out explicitly in sections 3 and 4 as a controlled iteration of a bilinear form estimate, with no circular appeal to stronger sum-product results or unverified solution counts. The stress-test note confirms the derivation tracks the threshold directly and recovers the product variant by substitution. That is real progress inside the finite-field covering literature.

The improvement shrinks as n grows, which is expected, and the two-dimensional near-half-density claim is stated without much elaboration in the abstract. Those are minor points rather than load-bearing issues. The Euclidean reduction follows the same numerical threshold without extra assumptions.

This is for people working on additive combinatorics, sum-product phenomena, and dimension problems. The explicit method and the clean improvement make it worth a serious referee's time even if revisions are needed on exposition or the two-dimensional part.

Referee Report

0 major / 2 minor

Summary. The manuscript proves finite-field covering results for the Peres-Schlag problem: for n≥3 and any η>0, |A|≫_{n,η} p^{3/(2n-1)+η} in F_p implies existence of t1,...,tn with t1A+⋯+tnA=F_p, improving the direct-product threshold 2/n. The same scale holds for the shifted-product variant (t1+A)⋯(tn+A). A two-dimensional near-half-density result is stated. In R, Borel A with dim_H A>2/n implies some shifted product contains a nonempty open interval. The proofs rely on an explicit additive-combinatorial argument via controlled iteration of a bilinear form estimate.

Significance. If the results hold, the work supplies a non-trivial improvement over the naive 2/n threshold by replacing it with 3/(2n-1) through an explicit bilinear-form iteration (detailed in §§3 and 4), together with a direct substitution for the shifted-product case and a dimension-reduction argument for the Euclidean statement. These explicit, non-circular arguments and the matching thresholds across linear, product, and real settings constitute a clear advance in additive combinatorics with potential implications for sum-product phenomena.

minor comments (2)

[Abstract] Abstract: the two-dimensional near-half-density result is announced without its precise exponent or statement; a one-sentence clarification would improve readability.
[§1] §1: the notation for the bilinear form estimate (used in the iteration) would benefit from an explicit displayed inequality with a forward reference to its first appearance in §3.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, accurate summary of the results, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives its improved exponent 3/(2n-1) via an explicit additive-combinatorial argument (Sections 3 and 4) that iterates a bilinear form estimate, replacing the naive direct-product threshold 2/n. The shifted-product variant follows by direct substitution of the same estimate, and the Euclidean statement by dimension reduction tracking the identical numerical threshold. No equation or claim reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the central claims remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard properties of finite fields and Hausdorff dimension; no free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math F_p is a field and linear combinations behave as expected under the field operations.
Invoked for the definition of t1A + ⋯ + tnA and the product (t1 + A)⋯(tn + A).
standard math Hausdorff dimension satisfies the standard monotonicity and countable-stability properties.
Used for the Euclidean statement that dim_H A > 2/n implies a shifted product contains an interval.

pith-pipeline@v0.9.1-grok · 5806 in / 1397 out tokens · 30500 ms · 2026-07-02T11:16:38.667163+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references

[1]

Bourgain, N

J. Bourgain, N. H. Katz and T. C. Tao,A sum-product estimate in finite fields, and applications, Geom. Funct. Anal.14(2004), no. 1, 27–57

2004
[2]

Chen,Projections in vector spaces over finite fields, Ann

C. Chen,Projections in vector spaces over finite fields, Ann. Acad. Sci. Fenn. Math.43(2018), no. 1, 171–185

2018
[3]

G. A. Edgar and C. L. Miller,Borel subrings of the reals, Proc. Amer. Math. Soc.131(2003), no. 4, 1121–1129

2003
[4]

K. J. Falconer, J. M. Fraser and X. Jin,Sixty years of fractal projections, in Fractal geometry and stochastics V, 3–25, Progr. Probab., 70, Birkh¨ auser/Springer, Cham
[5]

Hong, C.-W

G.-D. Hong, C.-W. Liang and C.-Y. Shen,On the Peres–Schlag orthogonal projection problem and Kakeya-type sets, preprint (2026)

2026
[6]

R. P. Kaufman,On Hausdorff dimension of projections, Mathematika15(1968), 153–155

1968
[7]

Mattila,Hausdorff dimension, orthogonal projections and intersections with planes, Ann

P. Mattila,Hausdorff dimension, orthogonal projections and intersections with planes, Ann. Acad. Sci. Fenn. Ser. A I Math.1(1975), no. 2, 227–244

1975
[8]

Mattila,Hausdorff dimension, projections, and the Fourier transform, Publ

P. Mattila,Hausdorff dimension, projections, and the Fourier transform, Publ. Mat.48(2004), no. 1, 3–4

2004
[9]

Mattila,Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, 150, Cambridge Univ

P. Mattila,Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, 150, Cambridge Univ. Press, Cambridge, 2015

2015
[10]

J. M. Marstrand,Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. London Math. Soc. (3)4(1954), 257–302

1954
[11]

Peres and W

Y. Peres and W. Schlag,Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions, Duke Math. J.102(2000), no. 2, 193–251

2000
[12]

E. M. Stein,Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series Monographs in Harmonic Analysis, 43 III, Princeton Univ. Press, Princeton, NJ, 1993

1993
[13]

Stevens and F

S. Stevens and F. de Zeeuw,An improved point-line incidence bound over arbitrary fields, Bull. Lond. Math. Soc.49(2017), no. 5, 842–858

2017
[14]

T. C. Tao and V. H. Vu,Additive combinatorics, Cambridge Studies in Advanced Mathematics, 105, Cambridge Univ. Press, Cambridge, 2006. Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA Email address:ghong@caltech.edu Department of Mathematics, National Taiwan University, Taiwan. Email address:d10221001@ntu.edu.tw Depar...

2006

[1] [1]

Bourgain, N

J. Bourgain, N. H. Katz and T. C. Tao,A sum-product estimate in finite fields, and applications, Geom. Funct. Anal.14(2004), no. 1, 27–57

2004

[2] [2]

Chen,Projections in vector spaces over finite fields, Ann

C. Chen,Projections in vector spaces over finite fields, Ann. Acad. Sci. Fenn. Math.43(2018), no. 1, 171–185

2018

[3] [3]

G. A. Edgar and C. L. Miller,Borel subrings of the reals, Proc. Amer. Math. Soc.131(2003), no. 4, 1121–1129

2003

[4] [4]

K. J. Falconer, J. M. Fraser and X. Jin,Sixty years of fractal projections, in Fractal geometry and stochastics V, 3–25, Progr. Probab., 70, Birkh¨ auser/Springer, Cham

[5] [5]

Hong, C.-W

G.-D. Hong, C.-W. Liang and C.-Y. Shen,On the Peres–Schlag orthogonal projection problem and Kakeya-type sets, preprint (2026)

2026

[6] [6]

R. P. Kaufman,On Hausdorff dimension of projections, Mathematika15(1968), 153–155

1968

[7] [7]

Mattila,Hausdorff dimension, orthogonal projections and intersections with planes, Ann

P. Mattila,Hausdorff dimension, orthogonal projections and intersections with planes, Ann. Acad. Sci. Fenn. Ser. A I Math.1(1975), no. 2, 227–244

1975

[8] [8]

Mattila,Hausdorff dimension, projections, and the Fourier transform, Publ

P. Mattila,Hausdorff dimension, projections, and the Fourier transform, Publ. Mat.48(2004), no. 1, 3–4

2004

[9] [9]

Mattila,Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, 150, Cambridge Univ

P. Mattila,Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, 150, Cambridge Univ. Press, Cambridge, 2015

2015

[10] [10]

J. M. Marstrand,Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. London Math. Soc. (3)4(1954), 257–302

1954

[11] [11]

Peres and W

Y. Peres and W. Schlag,Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions, Duke Math. J.102(2000), no. 2, 193–251

2000

[12] [12]

E. M. Stein,Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series Monographs in Harmonic Analysis, 43 III, Princeton Univ. Press, Princeton, NJ, 1993

1993

[13] [13]

Stevens and F

S. Stevens and F. de Zeeuw,An improved point-line incidence bound over arbitrary fields, Bull. Lond. Math. Soc.49(2017), no. 5, 842–858

2017

[14] [14]

T. C. Tao and V. H. Vu,Additive combinatorics, Cambridge Studies in Advanced Mathematics, 105, Cambridge Univ. Press, Cambridge, 2006. Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA Email address:ghong@caltech.edu Department of Mathematics, National Taiwan University, Taiwan. Email address:d10221001@ntu.edu.tw Depar...

2006