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

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 (η>0), this holds whenever [ |A|\gg_{n,η} p^{\frac{3}{2n-1}+η}. ] 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.

• The paper proves that for n ≥ 3, product sets Aⁿ over finite fields yield full dilated sumsets and shifted products when |A| ≳ p^(3/(2n−1)+η), improving the prior p^(2/n) scale.
• The paper uses additive or multiplicative energy averaging together with Cartesian-product incidence bounds to control complements and force one-dimensional images to cover the entire field.
• The paper proves that Euclidean shifted products contain an open interval when dim_H A > 2/n via logarithmic coordinates and Sobolev-regular convolution measures, while sharper dimension thresholds remain conjectural.

This paper studies finite-field and Euclidean analogues of the Peres–Schlag nonempty-interior problem for projections, specialized to sets with product structure. Given $A \subseteq \mathbb{F}_p$, the central question is when one can choose coefficients $t_1,\ldots,t_n$ such that the dilated sumset $t_1A + \cdots + t_nA$ equals all of $\mathbb{F}_p$ — equivalently, when a one-dimensional linear image of $A^n$ is full. The paper also develops a multiplicative variant in which linear forms are replaced by shifted products $(t_1+A)\cdots(t_n+A)$.

Background and motivation

The classical Marstrand–Mattila projection theorem states that for a Borel set $E \subseteq \mathbb{R}^n$ with $\dim_H E > k$, almost every $k$-dimensional projection has full dimension, and if $\dim_H E > k$ the projection has positive Lebesgue measure for almost every subspace. Peres and Schlag proved a stronger nonempty-interior statement at the threshold $t_1,\ldots,t_n$0 [1749437], which is sharp in general. Chen established the finite-field analogue [3753167]: if $t_1,\ldots,t_n$1 with $t_1,\ldots,t_n$2, then almost every $t_1,\ldots,t_n$3-dimensional projection of $t_1,\ldots,t_n$4 is full. The authors' own prior work improves this threshold to $t_1,\ldots,t_n$5, sharp by Kakeya-type constructions.

For product sets $t_1,\ldots,t_n$6, the Peres–Schlag–Chen threshold $t_1,\ldots,t_n$7 translates into the scale $t_1,\ldots,t_n$8. The main point of this paper is that product structure permits going below this scale: for $t_1,\ldots,t_n$9, the exponent $t_1A + \cdots + t_nA$0, which is strictly smaller than $t_1A + \cdots + t_nA$1, suffices.

Linear projections over finite fields

The first result addresses the two-dimensional near-half-density regime. If $t_1A + \cdots + t_nA$2, the Cauchy–Davenport inequality immediately gives $t_1A + \cdots + t_nA$3 for every $t_1A + \cdots + t_nA$4. The theorem shows that allowing the dilation $t_1A + \cdots + t_nA$5 to depend on $t_1A + \cdots + t_nA$6 extends this slightly below half density: there is an absolute $t_1A + \cdots + t_nA$7 (e.g., $t_1A + \cdots + t_nA$8) such that $t_1A + \cdots + t_nA$9 implies $\mathbb{F}_p$0 for some $\mathbb{F}_p$1. This is a fixed-constant improvement, not an exponent saving; whether $\mathbb{F}_p$2 suffices remains open.

The proof is a rigidity argument. Assuming $\mathbb{F}_p$3 for all $\mathbb{F}_p$4, each missing point $\mathbb{F}_p$5 yields an inclusion $\mathbb{F}_p$6, and intersecting two such sets shows the affine maps $\mathbb{F}_p$7 move $\mathbb{F}_p$8 by a small symmetric-difference defect $\mathbb{F}_p$9 with $A^n$0. The maps $A^n$1 are translations satisfying $A^n$2. If some $A^n$3 is a nontrivial translation, conjugating by all $A^n$4 produces all nonzero translations, and averaging $A^n$5 over $A^n$6 contradicts the density assumption. Otherwise all $A^n$7 are trivial, forcing the $A^n$8 to share a common fixed point; after translating, $A^n$9 for all $(t_1+A)\cdots(t_n+A)$0, and averaging over the multiplicative group again yields an incompatible inequality.

The higher-dimensional theorem is the main quantitative contribution: for $(t_1+A)\cdots(t_n+A)$1 and every $(t_1+A)\cdots(t_n+A)$2, if $(t_1+A)\cdots(t_n+A)$3, then $(t_1+A)\cdots(t_n+A)$4 for suitable nonzero coefficients. The proof proceeds in two stages. First, an averaging of the additive energy $(t_1+A)\cdots(t_n+A)$5 of the $(t_1+A)\cdots(t_n+A)$6-fold sum over $(t_1+A)\cdots(t_n+A)$7 — exploiting that each non-diagonal tuple determines at most one coefficient — produces an $(t_1+A)\cdots(t_n+A)$8-fold sumset $(t_1+A)\cdots(t_n+A)$9 whose complement $E \subseteq \mathbb{R}^n$0 satisfies $E \subseteq \mathbb{R}^n$1. Second, if $E \subseteq \mathbb{R}^n$2 never covers $E \subseteq \mathbb{R}^n$3, the inclusions $E \subseteq \mathbb{R}^n$4 generate $E \subseteq \mathbb{R}^n$5 lines each meeting $E \subseteq \mathbb{R}^n$6 in at least $E \subseteq \mathbb{R}^n$7 points; the Stevens–de Zeeuw Cartesian-product incidence estimate [3742451] then forces $E \subseteq \mathbb{R}^n$8, contradicting the complement bound. The hypothesis $E \subseteq \mathbb{R}^n$9 is exactly what makes $\dim_H E > k$0, so the incidence contradiction closes.

The finite-field exponent improvement motivates a Euclidean conjecture: for $\dim_H E > k$1 there exists $\dim_H E > k$2 such that $\dim_H E > k$3 implies some $\dim_H E > k$4 contains a nonempty open interval. This remains open in the paper.

Product-type projections over finite fields

Motivated by sum-product phenomena, the authors replace linear forms with shifted product maps. The two-fold near-half-density result parallels the linear one: if $\dim_H E > k$5, then $\dim_H E > k$6 for some shifts $\dim_H E > k$7. The proof handles the range $\dim_H E > k$8 directly via the multiplicative Cauchy–Davenport fact that $\dim_H E > k$9 whenever $k$0 with $k$1. In the remaining range, the rigidity scheme is multiplicative: the defect maps $k$2 now compose via $k$3, so the residual maps $k$4 are dilations. If some $k$5 is a nontrivial dilation, conjugation produces uniform bounds $k$6 for all translates, and averaging over $k$7 gives a contradiction. If all $k$8 are trivial, the group $k$9 is a subgroup of the affine group of order $\dim_H E > k$0; since any non-translation element has order dividing $\dim_H E > k$1 while every element of $\dim_H E > k$2 has order dividing $\dim_H E > k$3, all elements are translations, and averaging over translations again contradicts the density hypothesis.

The $\dim_H E > k$4-fold product theorem achieves the same exponent $\dim_H E > k$5: if $\dim_H E > k$6, then $\dim_H E > k$7 for suitable shifts. The proof mirrors the linear argument with two modifications. The first shift is chosen as $\dim_H E > k$8 so that $\dim_H E > k$9 lies in the partial product, separating off the zero issue. Two complementary averaging arguments control the nonzero complement $t_1,\ldots,t_n$00: averaging multiplicative energy over all shifts gives $t_1,\ldots,t_n$01, while restricting shifts to $t_1,\ldots,t_n$02 (so all but the first factor are nonzero) gives $t_1,\ldots,t_n$03; taking the better of the two yields $t_1,\ldots,t_n$04, which again suffices for the incidence contradiction against $t_1,\ldots,t_n$05 via the lines $t_1,\ldots,t_n$06.

Euclidean shifted products

The continuous result reaches the direct threshold $t_1,\ldots,t_n$07: for $t_1,\ldots,t_n$08 and Borel $t_1,\ldots,t_n$09 with $t_1,\ldots,t_n$10, some shifted product $t_1,\ldots,t_n$11 contains a nonempty open interval. The key idea is a logarithmic change of coordinates. Choosing all shifts large and positive, the pushforwards $t_1,\ldots,t_n$12 of a Frostman measure $t_1,\ldots,t_n$13 on a compact $t_1,\ldots,t_n$14 with finite $t_1,\ldots,t_n$15-energy, $t_1,\ldots,t_n$16, satisfy the averaged Fourier decay estimate

$t_1,\ldots,t_n$17

proved via van der Corput's lemma applied to the phase $t_1,\ldots,t_n$18, whose derivative is monotone and of size $t_1,\ldots,t_n$19. Since $t_1,\ldots,t_n$20, one can pick $t_1,\ldots,t_n$21 with $t_1,\ldots,t_n$22; Tonelli's theorem then yields shifts for which the convolution density $t_1,\ldots,t_n$23 lies in $t_1,\ldots,t_n$24, hence is continuous and nonnegative with total mass one. Its support therefore contains an interval, and exponentiating transfers this interval into the shifted product set.

The authors note that, in analogy with the finite-field results, the threshold $t_1,\ldots,t_n$25 is not expected to be optimal for the Euclidean product-type problem, and record the $t_1,\ldots,t_n$26-improvement as a conjecture.

Limitations and open questions

Several gaps between the proved results and the expected statements are acknowledged. The near-half-density theorems improve the trivial Cauchy–Davenport bound only by a fixed constant $t_1,\ldots,t_n$27, with no exponent saving; whether $t_1,\ldots,t_n$28 suffices for either $t_1,\ldots,t_n$29 or $t_1,\ldots,t_n$30 to cover $t_1,\ldots,t_n$31 is open. The Euclidean linear conjecture — that $t_1,\ldots,t_n$32 suffices for dilated sumsets to contain an interval — is stated but not proved, and its product-type counterpart (improving the proved threshold $t_1,\ldots,t_n$33) is likewise open. The finite-field results also require $t_1,\ldots,t_n$34 sufficiently large with constants depending on $t_1,\ldots,t_n$35 and $t_1,\ldots,t_n$36, and the incidence-based argument gives no information about the exact constant $t_1,\ldots,t_n$37.

Conclusion

The paper establishes that product structure lowers the density threshold for full one-dimensional images below the Peres–Schlag scale: over $t_1,\ldots,t_n$38, both dilated sums and shifted products of $t_1,\ldots,t_n$39 copies of $t_1,\ldots,t_n$40 cover the whole field whenever $t_1,\ldots,t_n$41, improving on the direct product analogue $t_1,\ldots,t_n$42 of the general projection threshold. In the Euclidean setting, the shifted-product problem is solved at the direct threshold $t_1,\ldots,t_n$43 via logarithmic coordinates and Sobolev regularity of convolutions. The uniform exponent $t_1,\ldots,t_n$44 across the additive and multiplicative settings suggests that the incidence-plus-energy mechanism, rather than the additive structure specifically, drives the improvement; closing the remaining gaps between the finite-field exponents, the Euclidean thresholds, and the conjectured $t_1,\ldots,t_n$45-improvements is the natural continuation of this work.