Given a compact abelian group $G$, endowed with a Haar probability measure $\mu$, let $\mathcal{F}_a$ be the set of (Borel) measurable functions $G\rightarrow [0,1]$ satisfying $\int_G fd\mu \geq \alpha$. The average of $f$ is called its density.

Let $\Psi$ be system of linear forms, i.e $\Psi=(\psi_1,\ldots,\psi_t): G^d\rightarrow G^t$ and

$\displaystyle{\psi_i(x_1,\ldots,x_d)=\sum_{j=1}^dc_{i,j}x_j}$

where $c_{i,j}\in\mathbb{Z}$ for any $i\in[t]$ and $j\in[d]$. For a measurable function $f : G\rightarrow [0,1]$, one may define the $\Psi$-count of $f$ as

$\displaystyle{T_\Psi(f)=\int_{G^d}\prod_i f(\psi_i(x))d\mu(x).}$

Here by a slight abuse of notation, the measure $\mu$ on $G^d$ denotes in fact the product measure $\mu^{\otimes d}$ of the Haar measure on $G$.

Let

$\displaystyle{m_\Psi(\alpha,G)=\inf_{f\in\mathcal{F}_a}T_\Psi(f).}$

When a system $\Psi$ is fixed, we may omit it from the notation $m_\Psi$. Given a sequence $G_n$ of finite groups of cardinality tending to infinity, it is natural to consider the sequence $m(\alpha,G_n)$. When the infimum is taken only over $\{0,1\}$-valued measurable functions of density at least $\alpha$, this is the minimum number of APs a set $A\subset G_n$ of density at least $\alpha$ can have. But these infimum are actually the same (basically by convexity).

The case $G_n=\mathbb{Z}/n\mathbb{Z}$ was introduced by Croot. He proved that the minimal density $m(\alpha,\mathbb{Z}_N)$ of three-term arithmetic progressions converged as $N\rightarrow\infty$ through the primes. Sisask, in his PhD thesis, was able to express this limit as an integral over the corresponding limit object, namely the circle $\mathbb{T}=\mathbb{R}/\mathbb{Z}$, thus

$\displaystyle{\lim_{\substack{N\rightarrow +\infty\\ N\text{ prime}}} m(\alpha,\mathbb{Z}_N)=m(\alpha,\mathbb{T}).}$

As customary, the corresponding problem in finite field models is cleaner. Thus we set $G_n=\mathbb{F}_p^n$. Whereas Croot noticed that the sequence $m(\alpha,\mathbb{Z}_N)$ was in some sense approximately decreasing, it is easy to see that for any system $\Psi$, the sequence $m_\Psi(\alpha,G_n)$ is genuinely non-increasing. This is because $\mathbb{F}_p^n$ embeds naturally in $\mathbb{F}_p^{n+1}$; thus if $f$ is a map from $\mathbb{F}_p^n$ to $[0,1]$, it can be extended into $\tilde{f} : \mathbb{F}_p^{n+1}\rightarrow [0,1]$ by putting $\tilde{f}(x_1,\ldots,x_{n+1})=f(x_1,\ldots,x_n)$.

Now $\mathbb{E} \tilde{f}=\mathbb{E}f$, and $T_\Psi(f)=T_\Psi(f)$. This proves that $m_\Psi(\alpha,G_n)\geq m_\Psi(\alpha,G_{n+1})$, which implies that this sequence has a limit.

Let $\Psi$ be now fixed. We want to find an appealing expression for the limit of the sequence $m(\alpha,G_n)$. It is thus tempting to look for a natural topological compact group that would play the role of limit object of the sequence $G_n$, just as $\mathbb{T}$ played the role of limit object for the sequence $\mathbb{Z}/N\mathbb{Z}$.

The analogy $\mathbb{Z}\leftrightarrow \mathbb{F}_p[t]=\cup \mathbb{F}_p^n=\mathbb{F}_p^{\infty}$ leads one to introducing $G_\mathbb{N}=\mathbb{F}_p^\mathbb{N}$ and $G_\infty=\mathbb{F}_p^{\infty}$. The group $G_\mathbb{N}$ equipped with the product topology is a compact group by Tychonoff’s theorem, and $G_\infty$ equipped with the discrete topology is locally compact; it is the Pontryagin dual of $G_\mathbb{N}$. It is now reasonable to ask whether

$\displaystyle{\lim m(\alpha,G_n)=m(\alpha, G_\mathbb{N}).}$

One easily notices that $m(\alpha,G_n)\geq m(\alpha, G_\mathbb{N})$ for any $n\in\mathbb{N}$ by extending, as above, a function $f : \mathbb{F}_p^n \rightarrow [0,1]$ into a function $\tilde{f}$ on $\mathbb{F}_p^\mathbb{N}$ with the same density and the same $\Psi$-count. Indeed, one can define

$\displaystyle{\tilde{f}((x_i)_{i\in\mathbb{N}})=f(x_1,\ldots,x_n)}$

and then $\tilde{f}$ is a measurable function which has same density and same $\Psi$-count.

We prove the other inequality. For this we prove two intermediate propositions. We write from henceforth $G=G_\mathbb{N}$.

Proposition 1

If $f_n$ is a sequence of $[0,1]$-valued functions on $G$ that converges in $L^2(G)$ to a $[0,1]$-valued function $f$, then $T(f_n)\rightarrow T(f)$.

Proof

Let $F_n(x)=\prod_{i\in[t]} f_n(\psi_i(x))-\prod_{i\in[t]} f(\psi_i(x))$. We show that $F_n$ converges in $L^1(G^d)$ to 0, which will conclude. In fact we prove a stronger convergence, a convergence in $L^2$. We show that $\lVert F_n-F\rVert_{2}\leq t \lVert f_n-f\rVert_2$. For that we simply remark that

$\displaystyle{F_n(x)=\sum_{j=1}^{t}\prod_{ij}f(\psi_k(x)).}$

Now, any non-trivial linear map $\theta : G^d\rightarrow G$, i.e. any map of the form

$\displaystyle{\theta(x_1,\ldots,x_d)=\sum_{i=1}^d c_ix_i.}$

with $(c_1,\ldots,c_d)\in \mathbb{F}_p^d\setminus\{0\}$, preserves the Haar measure, which implies that

$\displaystyle{\lVert f_n\circ\psi_j-f\circ\psi_j\rVert_{L^2(G^d)}=\lVert f_n-f\rVert_{L^2(G)}.}$

Using this, the triangle equality and the fact that the functions are 1-bounded, we conclude.

Proposition 2

Let $\mathcal{F}_{a,n}$ be the set of functions $\tilde{f}$ for $f : G_n\rightarrow [0,1]$ of density at least $\alpha.$ Then $\mathcal{F}_{a,n}$ is dense in $(\mathcal{F}_a, \lVert \cdot\rVert_2).$

Proof

Let $f\in\mathcal{F}_a$We construct a sequence of functions $f_n\in\mathcal{F}_{a,n}$ that converge to $f.$ We use duality and Fourier transform. We have $\hat{G_n}=G_n$ and $\hat{G_\mathbb{N}}=G_\infty=\cup G_n.$

For $\chi\in G_\infty$, write

Then write

This defines a function $f_n : G_n \rightarrow \mathbb{C}$, which obviously satisfies

.

The same holds for its extension $\tilde{f_n}$Notice that $\tilde{f_n}$ is also the conditional expectation with respect to the $\sigma$-algebra $\mathcal{P}(G_n)$More precisely,

$\displaystyle{f_n(x_1,\ldots,x_n)=\mathbb{E}_{y\in G}(f(y)\mid (y_1,\ldots,y_n)=(x_1,\ldots,x_n))}$

as both equal .

This implies in particular that $f_n$, and thus $\tilde{f_n}$has its values in $[0,1]$.

To check that $\mathbb{E} f_n \geq \alpha$, we use the conditional expectation interpretation

$\displaystyle{\mathbb{E}[f_n]=\mathbb{E}[\mathbb{E}[f\mid G_n]]=\mathbb{E}[f]\geq \alpha.}$

Now we check the convergence. It follows from basic harmonic analysis: $f$ being in $L^2(G)$, its Fourier series $f_n$ converges to $f$ in $L^2(G)$. This concludes the proof of Proposition 2.

Thus even for 4-APs, this shows that

$\displaystyle{\lim_{n\rightarrow\infty}\inf_{f:\mathbb{F}_p^n\rightarrow[0,1],\mathbb{E}f\geq\alpha}\mathbb{E}_{x,y}f(x)f(x+y)f(x+2y)f(x+3y)}$

exists and equals

$\displaystyle{\inf_{f:\mathbb{F}_p^\mathbb{N}\rightarrow[0,1],\int f\geq \alpha}\int f(x)f(x+y)f(x+2y)f(x+3y)dxdy.}$

This is in contrast to a paper of Candela-Szegedy, for the same pattern $\Psi$ (4-APs) but with $G_n=\mathbb{Z}/n\mathbb{Z}$ where they obtain as a limit a much more complicated infimum, an infimum over functions $f$ on $\mathbb{T}^2$ of the $\Psi'$-count, where $\Psi'\neq\Psi$ is another pattern. The part that holds in the setting of $\mathbb{F}_p^n$ and not in the setting of $\mathbb{Z}_p$ is Proposition 2. The truncated Fourier series of $f : \mathbb{T}\rightarrow[0,1]$, which is the natural $L^2$ approximation, does not give a function on cyclic groups $\mathbb{Z}_p$ with the same density and AP-count as $f$.

We remark that the paper of Hatami-Hatami-Hirst implies that the sequence of functions $f_n$ admits a limit object $g$ which is defined on a finite cartesian power of $G_\mathbb{N}$.

In fact, if $\Psi$ is the $k$-APs, pattern of complexity $k-2$, and $p\geq k$, the limit object $g$ is defined on $G_\mathbb{N}^{k-2}$. In particular, if $k=3$, this means that the infimum above is attained, i.e.

$\lim_{n\rightarrow \infty}m(\alpha, G_n)=T(g)$

for some $g : \mathbb{F}_p^\mathbb{N}\rightarrow [0,1]$ measurable. When , it is still not known whether such a result (i.e. the existence of a limit object $g : \mathbb{T}\rightarrow [0,1]$) holds. However, Szegedy proves the existence of a limit object $g : X\rightarrow [0,1]$ on a non-explicit group $X$.

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