on Green-Tao theorem

Overview

Green and Tao proved that the primes contain arbitrarily long arithmetic progressions (Green & Tao, 2008). More generally, every subset of the primes with positive relative upper density contains arbitrarily long arithmetic progressions. The proof does not prove new quantitative bounds in Szemeredi’s theorem, nor does it prove the Hardy-Littlewood prime tuples conjecture. Its central idea is different: replace the missing density of the primes by a pseudorandom measure which behaves like the uniform measure for the purpose of counting progressions.

The proof has three interacting parts. First, the $W$-trick removes local congruence obstructions. Second, a Goldston-Yildirim divisor-sum majorant places the primes inside a pseudorandom set of almost primes. Third, a relative Szemeredi theorem transfers the dense theorem from the uniform measure to that pseudorandom measure.

Statement and Density Form

An arithmetic progression of length $k$ is a set of the form

$$a,a+r,a+2r,\dots ,a+(k-1)r,$$

with $r\ne 0$. The theorem is:

Green-Tao theorem

For every $k\ge 3$, there are infinitely many arithmetic progressions of length $k$ consisting entirely of primes.

The paper proves a stronger density form:

Szemeredi theorem in the primes

If $A$ is a subset of the primes with positive relative upper density,

$$\underset{N\to \infty }{\mathrm{limsup}}\frac{|A\cap [1,N]|}{\pi (N)}>0,$$

then $A$ contains arbitrarily long arithmetic progressions.

The phrase “relative density” is important. The primes themselves have natural density zero in the integers:

$$\frac{|\mathcal{P}\cap [1,N]|}{N}\sim \frac{1}{\log N}.$$

Thus ordinary Szemeredi theory does not apply directly. Szemeredi’s theorem says that positive-density subsets of the integers contain long progressions (Szemeredi, 1975). The primes are far too sparse for this statement to see them.

The Green-Tao proof asks a different question. Suppose a sparse set is not arbitrary but is distributed like a random set at the level needed by the pattern-counting argument. Can dense combinatorics still apply inside it? The answer is yes, provided the sparse set has a sufficiently good pseudorandom majorant.

Progression Counts and Varnavides Averaging

The proof is organized around progression counts rather than isolated existence statements. Let $N$ be a large prime, and work in the cyclic group

$${\mathbb{Z}}_N=\mathbb{Z}/N\mathbb{Z}.$$

For a nonnegative function $f:{\mathbb{Z}}_N\to {\mathbb{R}}_{\ge 0}$, define the normalized $k$-term progression count

$${\Lambda }_k(f)={\mathbb{E}}_{x,r\in {\mathbb{Z}}_N}f(x)f(x+r)\cdots f(x+(k-1)r).$$

Here $\mathbb{E}$ denotes normalized average. If $f=1_A$, then ${\Lambda }_k(f)$ is the normalized number of cyclic $k$-term progressions in $A$.

The finitary counting form of Szemeredi’s theorem says:

Varnavides-Szemeredi counting form

For every $k\ge 3$ and $\delta >0$, there is a constant $c(k,\delta )>0$ such that every function $0\le f\le 1$ on ${\mathbb{Z}}_N$ with

$$\mathbb{E}f\ge \delta$$

satisfies

$${\Lambda }_k(f)\ge c(k,\delta )-o_{k,\delta }(1).$$

This is stronger than the existence of one progression. It follows from Szemeredi’s theorem by a Varnavides averaging argument (Varnavides, 1959): many long subprogressions inherit positive density, and Szemeredi’s theorem supplies a progression in a positive fraction of them.

Green and Tao need a version where the upper bound $f\le 1$ is replaced by

$$0\le f\le \nu ,$$

where $\nu$ is a sparse, unbounded, normalized measure. The measure $\nu$ should look uniform to the counting argument even though it is concentrated on almost primes.

Necessity of a Pseudorandom Majorant

The usual prime-counting weight is the von Mangoldt function

$$\Lambda (n)=\{\begin{array}{ll}\log p,& n=p^m,\\ 0,& \text{otherwise}.\end{array}$$

The prime number theorem says

$$\frac{1}{N}\sum _{n\le N}\Lambda (n)\to 1.$$

So $\Lambda$ is a normalized density for the primes. It is tempting to set $f=\Lambda$ and try to prove a weighted Szemeredi theorem directly.

This direct approach exceeds the available input. To show that $\Lambda$ itself is pseudorandom for all linear patterns of the type needed here would be close to the Hardy-Littlewood prime tuples conjecture. For example, a full asymptotic for

$${\mathbb{E}}_{n\le N}\Lambda (n+h_1)\cdots \Lambda (n+h_m)$$

for arbitrary fixed distinct shifts $h_i$ is essentially prime-tuple information. The Green-Tao theorem is far weaker than Hardy-Littlewood, so the proof must avoid requiring such information.

The key move is to majorize the primes by a larger object:

$$f\le \nu ,$$

where $\nu$ is not the prime weight itself but a sieve weight concentrated on almost primes. This almost-prime envelope is broad enough for available sieve estimates to prove pseudorandomness, while still narrow enough that the primes have positive relative density inside it.

This is the central transferable idea: when the target set is too arithmetic to be pseudorandom, embed it in a slightly larger pseudorandom universe.

Local Obstructions and the $W$-Trick

The primes are not pseudorandom modulo small primes. For instance, except for the prime $2$, no prime is even. Except for $3$, no prime is congruent to $0$ modulo $3$. These local biases are large and permanent.

Green and Tao remove them by passing to one residue class modulo

$$W=\prod _{p\le w}p,$$

where $w=w(N)$ tends to infinity very slowly. For simplicity one may focus on primes of the form

$$Wn+1.$$

On this progression, every small prime $p\le w$ has already been avoided. The paper uses the modified von Mangoldt function

$$\tilde{\Lambda }(n)=\{\begin{array}{ll}\frac{\phi (W)}{W}\log (Wn+1),& Wn+1\text{ is prime},\\ 0,& \text{otherwise}.\end{array}$$

This is the same as $\frac{\phi (W)}{W}\Lambda (Wn+1)$ on primes and deliberately discards higher prime powers, which do not matter for producing progressions of primes. The factor $\phi (W)/W$ renormalizes the average: Dirichlet’s theorem and the prime number theorem in arithmetic progressions imply that the average of $\tilde{\Lambda }$ is close to $1$ when $W$ grows slowly enough.

This restriction preserves the target theorem. If one finds a progression

$$n,n+r,\dots ,n+(k-1)r$$

for which all $W(n+jr)+1$ are prime, then

$$Wn+1,Wn+Wr+1,\dots ,Wn+(k-1)Wr+1$$

is a genuine progression of primes.

The role of the $W$-trick is structural. It separates two kinds of structure:

• local structure, coming from forced congruence obstructions modulo small primes;
• global randomness, represented by the expectation that after local obstructions are removed, the primes behave randomly enough for the desired linear patterns.

This separation is now a standard template in additive number theory.

Pseudorandom Measures

A measure in the Green-Tao paper is a nonnegative function

$$\nu :{\mathbb{Z}}_N\to {\mathbb{R}}_{\ge 0}$$

with normalized average

$$\mathbb{E}\nu =1+o(1).$$

It may be unbounded. This is essential: a sparse set of density about $1/\log N$ can be represented by a weight of size about $\log N$ on that sparse set and $0$ elsewhere.

For fixed progression length $k$, Green and Tao call $\nu$ $k$-pseudorandom if it satisfies a linear forms condition with parameters depending only on $k$, specifically $(k{2}^{k-1},3k-4,k)$ in the paper, together with a $2^{k-1}$-correlation condition. The exact constants are less important than the fact that they are fixed once $k$ is fixed. The two conditions have different roles.

Linear Forms Condition

Let ${\psi }_1,\dots ,{\psi }_m:{\mathbb{Z}}_N^t\to {\mathbb{Z}}_N$ be affine-linear forms

$${\psi }_i(x_1,\dots ,x_t)=\sum _{j=1}^t{L}_{ij}{x}_j+b_i,$$

with bounded rational coefficients, and assume no two coefficient vectors are rational multiples of one another. The linear forms condition requires

$${\mathbb{E}}_{x\in {\mathbb{Z}}_N^t}\prod _{i=1}^m\nu ({\psi }_i(x))=1+o(1)$$

for all systems of bounded complexity relevant to $k$-term progressions.

This says that distinct linear events sampled from $\nu$ behave independently. In a random model with $\nu$ equal to $\log N$ on a random set of density $1/\log N$, this condition would be true with high probability for fixed systems of nonparallel forms.

Correlation Condition

The linear forms condition does not control expressions like

$${\mathbb{E}}_x\nu (x+h_1)\nu (x+h_2)\cdots \nu (x+h_m)$$

when some shifts collide or have strong arithmetic relations. All the forms $x+h_i$ have the same linear part, so they are exactly the kind of parallel system excluded above.

The correlation condition gives an upper bound of the form

$${\mathbb{E}}_x\nu (x+h_1)\cdots \nu (x+h_m)\le \sum _{1\le i<j\le m}\tau (h_i-h_j),$$

where $\tau$ has bounded moments:

$${\mathbb{E}}_h\tau (h{)}^q=O_q(1)$$

for every fixed $q$. The precise formulation is technical, but the meaning is simple: correlations are allowed to spike at exceptional arithmetic differences, but those spikes must be rare.

This is designed for sieve weights. If $h$ has many prime divisors, then the events “almost prime at $n$” and “almost prime at $n+h$” may be more correlated than usual. The condition allows such exceptional $h$, provided their contribution is controlled in moments.

Relative Szemeredi Theorem

The main transference theorem is:

Relative Szemeredi theorem

Fix $k\ge 3$ and $\delta >0$. If $\nu$ is a $k$-pseudorandom measure on ${\mathbb{Z}}_N$ and

$$0\le f\le \nu ,\mathbb{E}f\ge \delta ,$$

then

$${\Lambda }_k(f)={\mathbb{E}}_{x,r}f(x)f(x+r)\cdots f(x+(k-1)r)\ge c(k,\delta )-o_{k,\delta }(1),$$

where $c(k,\delta )$ is the same positive constant coming from the dense counting form of Szemeredi’s theorem.

This theorem is the conceptual core of the paper. It says that, for counting $k$-term progressions, a pseudorandom sparse universe behaves like the whole group.

The proof has the same architecture as many later transference arguments:

1. decompose $f$ into a uniform part and a structured part;
2. show the uniform part does not affect the progression count;
3. show the structured part is bounded and has positive density;
4. apply the dense theorem to the structured part.

The technical difficulty is that $f$ is not bounded by $1$; it is only bounded by $\nu$, which can be as large as a power of $\log N$ or more. The pseudorandomness hypotheses are exactly what make the dense argument survive this unboundedness.

Gowers Uniformity Norms

Gowers introduced higher uniformity norms in his proof of Szemeredi’s theorem (Gowers, 2001). For $d\ge 1$, the $U^d$ norm on ${\mathbb{Z}}_N$ is defined by

$$\Vert g{\Vert }_{U^d}^{2^d}={\mathbb{E}}_{x,h_1,\dots ,h_d}\prod _{\omega \in \{0,1{\}}^d}g(x+\omega \cdot h),$$

for real-valued $g$. Here

$$\omega \cdot h={\omega }_1{h}_1+\cdots +{\omega }_d{h}_d.$$

The first cases are instructive:

$$\Vert g{\Vert }_{U^1}=|\mathbb{E}g|,$$

and $\Vert g{\Vert }_{U^2}$ measures Fourier uniformity. Higher $U^d$ norms measure higher-order additive structure.

The $U^{k-1}$ norm appears because the Green-Tao generalized von Neumann argument applies Cauchy-Schwarz repeatedly until the $k$-term progression average is dominated by a $(k-1)$-dimensional cube average. Modern true-complexity language gives sharper distinctions in some settings, but the proof here deliberately uses the robust $U^{k-1}$ control furnished by this Cauchy-Schwarz scheme.

Green and Tao prove a relative generalized von Neumann theorem:

Relative generalized von Neumann principle

Suppose $\nu$ is $k$-pseudorandom and $f_0,\dots ,f_{k-1}$ are bounded pointwise by $\nu +1$. Then the multilinear progression average

$${\mathbb{E}}_{x,r}{f}_0(x)f_1(x+r)\cdots f_{k-1}(x+(k-1)r)$$

is controlled by the smallest relevant $U^{k-1}$ norm of the $f_i$, up to $o(1)$ errors.

The proof is a Cauchy-Schwarz machine. Each Cauchy-Schwarz step doubles the number of factors and introduces new affine-linear forms. At the end one obtains averages of products of $\nu$ along a finite system of linear forms. The linear forms condition says those averages are $1+o(1)$, so the usual dense Cauchy-Schwarz argument still works.

This is one of the lessons of the paper: the pseudorandomness conditions are not arbitrary. They are exactly the conditions generated by the proof after repeatedly applying Cauchy-Schwarz.

Uniform-Structured Decomposition

In the dense setting, one often decomposes

$$f=f_{\mathrm{str}}+f_{\mathrm{unf}},$$

where $f_{\mathrm{unf}}$ has small Gowers norm and $f_{\mathrm{str}}$ is controlled by a bounded-complexity factor.

Green and Tao implement a finitary ergodic-theoretic version. The structured object is a conditional expectation

$$\mathbb{E}(f\mid \mathcal{B})$$

onto a finite sigma-algebra $\mathcal{B}$. The uniform part is

$$f-\mathbb{E}(f\mid \mathcal{B}).$$

The desired conclusion is

$$\Vert f-\mathbb{E}(f\mid \mathcal{B}){\Vert }_{U^{k-1}}\text{ is small}.$$

There is a complication: because $f$ is only bounded by $\nu$, the conditional expectation might not be bounded by $1$. The proof constructs an exceptional set $\Omega$ such that

$$\mathbb{E}(\nu 1_{\Omega })=o(1),$$

and outside $\Omega$ one has

$$(1-1_{\Omega })\mathbb{E}(\nu -1\mid \mathcal{B})=o(1)$$

in $L^{\infty }$. Thus, outside a negligible bad set, the measure $\nu$ looks like the uniform measure when observed through the structured factor $\mathcal{B}$.

Define the bounded structured approximation

$$f_{\mathrm{str}}=(1-1_{\Omega })\mathbb{E}(f\mid \mathcal{B}).$$

Then $f_{\mathrm{str}}$ is bounded by $1+o(1)$ and has essentially the same mean as $f$. The dense Szemeredi theorem applies to $f_{\mathrm{str}}$. The difference between the progression counts of $f$ and $f_{\mathrm{str}}$ is controlled by the generalized von Neumann theorem because the error has small $U^{k-1}$ norm.

This is the transfer in operational form:

$$\text{sparse count for }f\approx \text{dense count for }{f}_{\mathrm{str}}>0.$$

Sieve Majorant

It remains to construct a pseudorandom $\nu$ that majorizes the $W$-tricked primes. This is where the number theory enters.

For a parameter $R$, define the truncated divisor sum

$${\Lambda }_R(n)=\sum _{\begin{array}{c}d\mid n\\ d\le R\end{array}}\mu (d)\log (R/d).$$

If $n$ is prime and $n>R$, then the only divisor $d\le R$ contributing in the sum is $d=1$, so

$${\Lambda }_R(n)=\log R.$$

Thus ${\Lambda }_R(n{)}^2/\log R$ is a nonnegative weight which is large on primes and whose average behavior is governed by the absence or presence of small divisors. It is not literally supported only on rough numbers; rather, it is a Selberg-sieve type majorant whose correlations can be estimated by expanding the divisor sums.

Green and Tao use a localized weight of the form

$$\nu (n)\approx \frac{\phi (W)}{W}\frac{{\Lambda }_R(Wn+1{)}^2}{\log R}$$

on a carefully chosen interval inside ${\mathbb{Z}}_N$, and set $\nu (n)=1$ outside that interval. The interval localization prevents wraparound progressions in ${\mathbb{Z}}_N$ from becoming fake progressions in the integers.

The paper takes $R=N^{k^{-1}{2}^{-k-4}}$. This small power is large enough that $\log R$ is a fixed positive multiple of $\log N$, so the majorant dominates the modified prime weight up to a constant depending only on $k$. It is also small enough that the Goldston-Yildirim estimates apply uniformly to all linear systems required by the relative theorem.

Goldston-Yildirim Correlation Estimates

The decisive input is an asymptotic for averages of products of these truncated divisor sums along affine-linear forms:

$${\mathbb{E}}_x\prod _{i=1}^m{\Lambda }_R({\psi }_i(x){)}^2.$$

For nonparallel forms, the answer factors as expected, up to local factors. After the $W$-trick removes small-prime irregularities, the local factors are close enough to $1$ to give the linear forms condition.

At a formal level, expand

$${\Lambda }_R(n{)}^2=\sum _{\begin{array}{c}d,d^{\prime }\le R\\ d\mid n,d^{\prime }\mid n\end{array}}\mu (d)\mu (d^{\prime })\log (R/d)\log (R/d^{\prime }).$$

Then a product over $i=1,\dots ,m$ becomes a large divisor sum. The average over $x$ asks for simultaneous divisibility conditions

$$[d_i,d_i^{\prime }]\mid {\psi }_i(x).$$

For independent linear forms, the number of solutions modulo the combined modulus approximately factors. This produces an Euler product.

The analytic work is to evaluate that Euler product accurately enough. Green and Tao follow Goldston and Yildirim’s contour-integral method (Goldston & Yildirim, 2007). The zeta function enters because the divisor sums are encoded by Dirichlet series. A classical zero-free region for $\zeta (s)$ allows the contours to be shifted with acceptable error.

For the correlation condition, the forms are parallel:

$$x+h_1,x+h_2,\dots ,x+h_m.$$

One no longer expects a clean $1+o(1)$ asymptotic uniformly in the shifts. If several differences $h_i-h_j$ have many prime divisors, the local factors can be large. The proof packages these local excesses into the function $\tau (h)$ with bounded moments. This proves the correlation condition rather than full independence.

The majorant therefore has exactly the two features needed by the transfer theorem:

• nonparallel linear patterns see it as uniform;
• parallel correlations may spike, but only on a controlled exceptional set of differences.

Assembly of the Proof

Fix $k$. Choose $w=w(N)$ slowly increasing and set

$$W=\prod _{p\le w}p.$$

Choose a large prime $N$ and identify an interval inside ${\mathbb{Z}}_N$ with an interval of integers. Define the $W$-tricked prime weight

$$\tilde{\Lambda }(n)=\frac{\phi (W)}{W}\Lambda (Wn+1).$$

After localization and multiplication by a constant depending only on $k$, this weight is bounded above by the sieve majorant $\nu$:

$$0\le f(n)\le \nu (n).$$

The average of $f$ over the chosen interval is bounded below by a positive constant depending on $k$, because the prime number theorem in arithmetic progressions gives the right average for $\tilde{\Lambda }$.

The Goldston-Yildirim estimates show that $\nu$ is $k$-pseudorandom. The relative Szemeredi theorem then gives

$${\mathbb{E}}_{x,r}f(x)f(x+r)\cdots f(x+(k-1)r)>0$$

for all sufficiently large $N$.

The contribution of the degenerate case $r=0$ is negligible. The interval localization ensures that nondegenerate cyclic progressions correspond to genuine integer progressions. Since $f$ is supported where $Wn+1$ is prime, a positive weighted count gives a genuine $k$-term progression of primes.

The same argument proves the relative-density form. If $A$ is a positive-relative-density subset of the primes, one chooses a residue class $b(\mathrm{mod}W)$ in which $A$ retains positive relative density and applies the same argument to $Wn+b$.

Scope of the Theorem

The theorem is qualitative. It gives a positive lower bound of the form

$$(\gamma (k)+o(1))\frac{N^2}{{\log }^{k}N}$$

for some extremely small $\gamma (k)>0$, but not the Hardy-Littlewood asymptotic. The constant is poor because the proof depends on qualitative or very inefficient quantitative forms of Szemeredi’s theorem and on slowly decaying pseudorandomness errors.

The proof also does not show that prescribed prime constellations occur. It finds progressions, a pattern already forced in every dense subset of the integers. The transfer principle can only transfer patterns that are known in dense sets and for which the sparse majorant satisfies the required pseudorandomness conditions.

Finally, the proof does not directly show that the primes themselves are pseudorandom. It shows that the primes are dense inside a pseudorandom majorant. This distinction is crucial. Proving the needed pseudorandomness for the prime weight itself would be much closer to Hardy-Littlewood.

Transferable Mechanisms

The Green-Tao theorem became a template for later transference arguments because its proof separates the source of density, the source of randomness, and the dense theorem being transferred. Later expositions and refinements make this separation explicit, especially the Conlon-Fox-Zhao exposition and the subsequent relative Szemeredi theorems (Conlon, Fox & Zhao, 2014; Conlon, Fox & Zhao, 2015; Zhao, 2014).

The first reusable mechanism is local normalization before pseudorandomness. The $W$-trick removes deterministic congruence obstructions before the measure is asked to satisfy uniformity conditions. This same local-normalization principle appears in later prime-pattern problems, including polynomial progressions in the primes and Gaussian-prime constellations (Tao & Ziegler, 2008; Tao, 2006).

The second mechanism is majorization followed by transfer. The proof does not require the von Mangoldt weight itself to satisfy the full pseudorandomness axioms. It constructs a larger almost-prime measure, proves pseudorandomness there, and then uses positive relative density inside that measure. Conlon, Fox, and Zhao later showed that the relative Szemeredi conclusion can be obtained under weaker pseudorandomness hypotheses, clarifying which parts of the original majorant structure are essential (Conlon, Fox & Zhao, 2015).

The third mechanism is proof-generated pseudorandomness. The linear forms condition is not an external guess: it is generated by the affine-linear systems created by repeated Cauchy-Schwarz in the relative generalized von Neumann theorem. The correlation condition handles the exceptional parallel configurations where pure linear-forms independence is not expected. Zhao’s arithmetic transference proof isolates this point especially clearly by using a dense model theorem, a counting lemma, and Szemeredi’s theorem as separate inputs (Zhao, 2014).

The fourth mechanism is uniform-structured decomposition under a sparse envelope. The $U^{k-1}$-uniform component is invisible to $k$-term progression counts, while the structured component must be converted into a bounded dense object before Szemeredi’s theorem can be invoked. This is the same organizing philosophy behind Gowers’s proof of Szemeredi’s theorem, but Green and Tao apply it relative to an unbounded pseudorandom measure (Gowers, 2001; Green & Tao, 2008).

The fifth mechanism is qualitative transference in place of unattainable quantitative density. A direct quantitative application of Szemeredi’s theorem to the primes would require far stronger bounds than are available. Green and Tao instead change the ambient measure. Later refinements show that this was not merely a technical detour: relative Szemeredi theorems form a robust bridge from dense additive combinatorics to sparse pseudorandom settings (Conlon, Fox & Zhao, 2014; Conlon, Fox & Zhao, 2015).

In compressed form, the argument is:

$$\text{primes}\subset \text{pseudorandom almost primes}\approx \text{dense integers}.$$

The proof succeeds because each arrow is proved by a different method: sieve majorization for the first inclusion, Goldston-Yildirim estimates for pseudorandomness, and relative Szemeredi transference for the final dense comparison.

See Also

• on bounded gaps between primes: Uses the same divisor-sum and sieve-weight vocabulary, but with a different objective: producing prime-rich admissible tuples rather than transferring a dense theorem.
• on sum-product in finite fields via entropy: Provides a contrasting model of pseudorandomness in finite fields, where additive and multiplicative structure are forced to separate.
• on lower bounds for incidences: Shares the theme that the useful hypotheses are often dictated by the proof mechanism rather than by an abstract randomness definition.

References

🐻  Conlon, D., Fox, J. & Zhao, Y. 2014. The Green-Tao theorem: an exposition. EMS Surveys in Mathematical Sciences 1(2), 249–282.

🐻  Conlon, D., Fox, J. & Zhao, Y. 2015. A relative Szemeredi theorem. Geometric and Functional Analysis 25(3), 733–762.

🐻  Goldston, D.A. & Yildirim, C.Y. 2007. Higher correlations of divisor sums related to primes. III: Small gaps between primes. Annales de l’Institut Fourier 57(4), 1377–1432.

🐻  Gowers, W.T. 2001. A new proof of Szemeredi’s theorem. Geometric and Functional Analysis 11(3), 465–588.

🐻  Green, B. & Tao, T. 2008. The primes contain arbitrarily long arithmetic progressions. Annals of Mathematics 167(2), 481–547.

🐻  Szemeredi, E. 1975. On sets of integers containing no k elements in arithmetic progression. Acta Arithmetica 27, 199–245.

🐻  Tao, T. 2006. The Gaussian primes contain arbitrarily shaped constellations. Journal d’Analyse Mathématique 99, 109–176.

🐻  Tao, T. & Ziegler, T. 2008. The primes contain arbitrarily long polynomial progressions. Acta Mathematica 201(2), 213–305.

🐻  Varnavides, P. 1959. On certain sets of positive density. Proceedings of the London Mathematical Society 9(3), 531–540.

🐻  Zhao, Y. 2014. An arithmetic transference proof of a relative Szemeredi theorem. Mathematical Proceedings of the Cambridge Philosophical Society 156(2), 255–261.