Arithmetic regularity as an alternative to transference

Abstract: Since Green (2005), the Fourier-analytic transference principle has dominated the landscape of combinatorial theorems relative to sparse arithmetic sets. We demonstrate a different approach using arithmetic regularity. This is more versatile and has the potential to succeed when no obvious dense model' is forthcoming. Moreover, we contend that, just as the traditional circle method disassembles an arithmetic problem into real and $p$-adic parts which can be solved individually, the arithmetic regularity method generalises this to yield an additionalcombinatorial' factor. This framework leads directly to a correct lower bound on the number of configurations in a dense set. We illustrate this using a system comprising a linear equation together with a higher-degree equation.

• The paper develops an arithmetic regularity framework that avoids constructing an ad hoc dense model, combining polynomial-phase decompositions, quantitative circle-method estimates, equidistribution, and compact counting.
• The paper proves that for s ≥ s₀(k), every dense subset A of [N] contains ≫δ N^(s−k−1) solutions to a linear-plus-degree-k system under non-proportionality or suitable smooth adelic hypotheses.
• The paper establishes qualitatively new existence results for k ≥ 3 and identifies current limitations imposed by Vinogradov mean value and restriction estimates, including the unresolved ten-variable cubic case.

Overview

The paper by Chow, Prendiville, and Vazquez develops an alternative to the Fourier-analytic transference principle for establishing Roth-type theorems relative to sparse arithmetic sets. Since Green's 2005 work on Roth's theorem in the primes [(1609.1609)-style results, see Gre2005], transference has dominated this area, but it requires constructing an ad-hoc "dense model problem" whose solutions are then transferred to the sparse setting. The authors argue that an arithmetic regularity approach removes this design step: applying the regularity lemma naturally outputs the appropriate dense problem. They illustrate the method on a system comprising a linear equation together with a degree-$k$ equation, obtaining a sharp supersaturation bound in dense variables — a result that is new even qualitatively for $k \geqslant 3$.

The main theorem

Let $k \geqslant 2$ and $s \geqslant s_0(k)$, where $s_0$ is tabulated for $2 \leqslant k \leqslant 6$ (e.g. $s_0(2)=7$, $s_0(3)=11$, $s_0(4)=17$) and given for $k \geqslant 7$ by

$k \geqslant 3$0

with $k \geqslant 3$1 determined by whether $k \geqslant 3$2 equals $k \geqslant 3$3. Let $k \geqslant 3$4 with $k \geqslant 3$5, and let $k \geqslant 3$6 have density $k \geqslant 3$7. Under either of two hypotheses — (A) the vectors $k \geqslant 3$8 are not proportional, or (B) they are proportional and the associated variety has smooth points over $k \geqslant 3$9 and every $k \geqslant 2$0 — the paper proves

$k \geqslant 2$1

where $k \geqslant 2$2 is defined by $k \geqslant 2$3. This is a supersaturation statement: it gives the correct-order lower bound on the number of solutions in a dense set, not merely existence. For $k \geqslant 2$4 the existence of even one non-trivial solution over $k \geqslant 2$5 was previously unknown except in special cases; for $k \geqslant 2$6 in Case B, Keil had established existence but not the lower bound $k \geqslant 2$7.

The hypotheses matter: when $k \geqslant 2$8 with mixed signs, Jensen-type obstructions can force all real solutions onto the diagonal, where the variety is singular, so no density result is possible.

The threshold $k \geqslant 2$9 arises from Vinogradov mean value estimates: the moment bounds used require $s \geqslant s_0(k)$0 variables at exponent $s \geqslant s_0(k)$1, drawing on Wooley's nested efficient congruencing and the Bourgain–Demeter–Guth resolution of the main conjecture. The authors note that for $s \geqslant s_0(k)$2, Wooley's odd-cubic Weyl sum estimates suggest ten variables might suffice, but the corresponding restriction estimate is unavailable below that exponent, so $s \geqslant s_0(k)$3 remains out of reach.

Arithmetic regularity for polynomial phases

The methodological core is Lemma ARLgen, an arithmetic regularity lemma adapted to polynomial phases. Given $s \geqslant s_0(k)$4 of density $s \geqslant s_0(k)$5 and polynomials $s \geqslant s_0(k)$6 of degrees $s \geqslant s_0(k)$7, it produces:

• a decomposition $s \geqslant s_0(k)$8 with $s \geqslant s_0(k)$9;
• an error term $s_0$0 with small $s_0$1-seminorm, where $s_0$2 and $s_0$3 is the span of the phases;
• a structured part $s_0$4 supported on an arithmetic progression $s_0$5 with $s_0$6 and $s_0$7 control inherited from $s_0$8;
• a trigonometric polynomial $s_0$9 of bounded degree such that $2 \leqslant k \leqslant 6$0 on $2 \leqslant k \leqslant 6$1.

The proof follows Tao's energy-increment strategy but introduces a technical device the authors call random translate Bohr neighbourhoods: instead of regular Bohr sets, functions are projected onto atoms of randomly translated inhomogeneous Bohr partitions. This simplifies several calculations — notably, a random-translate inverse theorem and a refined weak regularity lemma amenable to iteration yield full regularity with both $2 \leqslant k \leqslant 6$2-control and $2 \leqslant k \leqslant 6$3 closeness between successive projections. A rationality decomposition (in the spirit of Ratner/Tao) then extracts $2 \leqslant k \leqslant 6$4-irrational frequencies from the projection, and Fejér-kernel convolution converts the Lipschitz structure into a trigonometric polynomial.

The choice of seminorm is calibrated: the standard $2 \leqslant k \leqslant 6$5 regularity lemma gives ample information if $2 \leqslant k \leqslant 6$6 but weak information about the structured part, while $2 \leqslant k \leqslant 6$7 only prevents correlation with linear phases. The $2 \leqslant k \leqslant 6$8-seminorm against the specific phase family $2 \leqslant k \leqslant 6$9 provides exactly the uniformity needed.

Quantitative weak approximation via the circle method

Two counting tasks require precise dependence on the progression $s_0(2)=7$0, handled by quantitative versions of the Hardy–Littlewood method. First, the paper establishes asymptotics for $s_0(2)=7$1, the number of solutions to the system within $s_0(2)=7$2, of shape

$s_0(2)=7$3

with singular series $s_0(2)=7$4 (Case A) or $s_0(2)=7$5 (Case B) and matching singular integral lower bounds. The minor arc analysis combines Weyl-type major arc detection (via Baker's Diophantine approximation) with the moment estimate $s_0(2)=7$6, which for $s_0(2)=7$7 uses a careful reading of Wooley's efficient congruencing yielding an improvement over a direct application of his Corollary 14.8 in the equality case $s_0(2)=7$8.

Second, an auxiliary system in $s_0(2)=7$9 variables (counting pairs of solutions sharing a coordinate) is shown to satisfy $s_0(3)=11$0 (Case A) or $s_0(3)=11$1 (Case B), giving the key relation $s_0(3)=11$2. The $s_0(3)=11$3-adic analysis splits according to whether $s_0(3)=11$4 exceeds a constant depending on the variety, using Hensel lifting near smooth points and explicit evaluation of exponential sums at low $s_0(3)=11$5-adic scales. In Case B with large $s_0(3)=11$6, the analysis reduces to a quadratic system modulo $s_0(3)=11$7, exploiting the expansion of $s_0(3)=11$8 around the diagonal.

Configuration control and equidistribution

Two generalised von Neumann lemmas show the error terms contribute negligibly. Fourier control uses Hölder interpolation between the moment bound and a restriction estimate of Henriot–Hughes type (Bourgain's restriction bound for the curve $s_0(3)=11$9 when $s_0(4)=17$0): if $s_0(4)=17$1 then some $s_0(4)=17$2. $s_0(4)=17$3 control, proved by Cauchy–Schwarz combined with the auxiliary system bound, gives $s_0(4)=17$4.

Equidistribution exploits the irrationality of the regularity lemma's frequencies. A Weyl-type lemma shows that a large exponential sum forces rationality of the coefficients; contrapositively, irrational frequencies give power-saving bounds on individual sums. Expanding the structured count into Fourier series, all non-degenerate characters are killed by equidistribution, leaving only characters compatible with the linear relations among the coefficient vectors. The surviving integral is over the compact groups $s_0(4)=17$5, converting the finite count into a torus density.

Assembling the proof

The final step applies a compact Szemerédi result — here elementary by Fourier analysis on tori, though in general requiring hypergraph regularity — to the trigonometric polynomial $s_0(4)=17$6: since $s_0(4)=17$7, the constrained product integral is bounded below by $s_0(4)=17$8. Chaining the inequalities,

$s_0(4)=17$9

and $k \geqslant 7$0 (using the quantitative weak approximation lower bound $k \geqslant 7$1 to absorb error terms), yields $k \geqslant 7$2. Notably, the framework delivers supersaturation directly, without inheriting it from a different system as transference arguments must.

Limitations and open questions

The paper concedes several restrictions. The variable count $k \geqslant 7$3 is dictated by available moment and restriction estimates rather than by any intrinsic obstruction; in particular, the case $k \geqslant 7$4 for $k \geqslant 7$5 fails because the discrete restriction theory for $k \geqslant 7$6 is insufficiently developed. Case B requires smooth adelic points on the diagonal variety, and the Jensen inequality example shows the theorem cannot hold without some non-singularity hypothesis. The random translate Bohr machinery, while simplifying calculations, is not essential to the application, and its broader utility is asserted rather than demonstrated. Whether the arithmetic regularity approach can reach sparser sets or systems where no dense model problem is apparent — the motivating ambition — is tested here only on a single system.

Conclusion

This paper establishes a correct-order supersaturation bound for solutions to a linear-plus-degree-$k \geqslant 7$7 system inside dense sets, proving qualitative novelty for $k \geqslant 7$8, and packages the argument as a reusable template: arithmetic regularity with polynomial-phase seminorms, quantitative circle method with tracked progression dependence, equidistribution of irrational frequencies, and a compact counting step. The framework generalises the classical decomposition of the circle method into archimedean and $k \geqslant 7$9-adic factors by adding a combinatorial factor, and offers a systematic substitute for the dense-model construction that transference requires.