7 The integer case

Theorem 7.1

There is a constant \(c{\gt}0\) such that the following holds. Let \(\epsilon {\gt}0\) and \(B,B'\subseteq G\) be regular Bohr sets of rank \(d\). Suppose that \(A_1\subseteq B\) with density \(\alpha _1\) and \(A_2\) is such that there exists \(x\) with \(A_2\subseteq B'-x\) with density \(\alpha _2\). Let \(S\) be any set with \(\left\lvert S\right\rvert \leq 2\left\lvert B\right\rvert \). There is a regular Bohr set \(B''\subseteq B'\) of rank at most

\[ d+O_\epsilon (\mathcal{L}{\alpha _1}^3\mathcal{L}{\alpha _2}) \]

and size

\[ \left\lvert B''\right\rvert \geq \exp (-O_\epsilon (d\mathcal{L}{\alpha _1\alpha _2/d}+\mathcal{L}{\alpha _1}^3\mathcal{L}{\alpha _2}\mathcal{L}{\alpha _1\alpha _2/d}))\left\lvert B'\right\rvert \]

such that

\[ \left\lvert \langle \mu _{B'}\ast \mu _{A_1}\circ \mu _{A_2},1_{S}\rangle -\langle \mu _{A_1}\circ \mu _{A_2},1_{S}\rangle \right\rvert \leq \epsilon . \]

Proof ▶

To do.

Proposition 7.2

There is a constant \(c{\gt}0\) such that the following holds. Let \(\epsilon {\gt}0\) and \(p \geq 2\) be an integer. Let \(B \subseteq G\) be a regular Bohr set and \(A\subseteq B\) with relative density \(\alpha \). Let \(\nu : G \to \mathbb {R}_{\geq 0}\) be supported on \(B_\rho \), where \(\rho \leq c\epsilon \alpha /\operatorname{rank}(B)\), such that \(\lVert \nu \rVert _1=1\) and \(\widehat{\nu }\geq 0\). If

\[ \lVert (\mu _A-\mu _B) \circ (\mu _{A}-\mu _B)\rVert _{p(\nu )} \geq \epsilon \, \mu (B)^{-1}, \]

then there exists \(p'\ll _\epsilon p\) such that

\[ \lVert \mu _{A}\circ \mu _{A}\rVert _{p'(\nu )} \geq \left(1+\tfrac {1}{4}\epsilon \right) \mu (B)^{-1}. \]

Proof ▶

To do.

Proposition 7.3

There is a constant \(c{\gt}0\) such that the following holds. Let \(p \geq 2\) be an even integer. Let \(f : G \to \mathbb {R}\), let \(B \subseteq G\) and \(B', B'' \subseteq B_{c/\operatorname{rank}(B)}\) all be regular Bohr sets. Then

\[ \lVert f\circ f \rVert _{p(\mu _{B'}\circ \mu _{B'}\ast \mu _{B''}\circ \mu _{B''})} \geq \tfrac {1}{2} \lVert f*f\rVert _{p(\mu _B)}. \]

Proof ▶

To do,

Proposition 7.4

There is a constant \(c{\gt}0\) such that the following holds. Let \(\epsilon {\gt}0\). Let \(B \subseteq G\) be a regular Bohr set and \(A\subseteq B\) with relative density \(\alpha \), and let \(B' \subseteq B_{c\epsilon \alpha /\operatorname{rank}(B)}\) be a regular Bohr set and \(C\subseteq B'\) with relative density \(\gamma \). Either

\(\left\lvert \langle \mu _A*\mu _A, \mu _{C} \rangle - \mu (B)^{-1} \right\rvert \leq \epsilon \mu (B)^{-1}\) or
there is some \(p \ll \mathcal{L}{\gamma }\) such that \(\lVert (\mu _A-\mu _B)*(\mu _A-\mu _B)\rVert _{p(\mu _{B'})} \geq \tfrac {1}{2}\epsilon \mu (B)^{-1}\).

Proof ▶

To do.

Proposition 7.5

There is a constant \(c{\gt}0\) such that the following holds. Let \(\epsilon {\gt}0\) and \(p,k\geq 1\) be integers such that \((k,\left\lvert G\right\rvert )=1\). Let \(B,B',B''\subseteq G\) be regular Bohr sets of rank \(d\) such that \(B''\subseteq B'_{c/d}\) and \(A\subseteq B\) with relative density \(\alpha \). If

\[ \lVert \mu _{A}\circ \mu _{A}\rVert _{p(\mu _{k\cdot B'}\circ \mu _{k\cdot B'}\ast \mu _{k\cdot B''}\circ \mu _{k\cdot B''})} \geq \left(1+\epsilon \right) \mu (B)^{-1}, \]

then there is a regular Bohr set \(B'''\subseteq B''\) of rank at most

\[ \operatorname{rank}(B''')\leq d+O_{\epsilon }(\mathcal{L}{\alpha }^4p^4) \]

and size

\[ \left\lvert B'''\right\rvert \geq \exp (-O_{\epsilon }(dp\mathcal{L}{\alpha /d}+\mathcal{L}{\alpha }^5p^5))\left\lvert B''\right\rvert \]

such that

\[ \lVert \mu _{B'''}*\mu _A \rVert _\infty \geq (1+c\epsilon )\mu (B)^{-1}. \]

Proof ▶

To do.

Theorem 7.6

There is a constant \(c{\gt}0\) such that the following holds. Let \({\epsilon ,\delta \in (0,1)}\) and \(p,k\geq 1\) be integers such that \((k,\left\lvert G\right\rvert )=1\). For any \(A\subseteq G\) with density \(\alpha \) there is a regular Bohr set \(B\) with

\[ d=\operatorname{rank}(B) =O_{\epsilon }\left(\mathcal{L}{\alpha }^5p^4\right) \quad \text{and}\quad \left\lvert B\right\rvert \geq \exp \left(-O_{\epsilon ,\delta }(\mathcal{L}{\alpha }^6p^5\mathcal{L}{\alpha /p})\right)\left\lvert G\right\rvert \]

and some \(A'\subseteq (A-x)\cap B\) for some \(x \in G\) such that

\(\left\lvert A'\right\rvert \geq (1-\epsilon )\alpha \left\lvert B\right\rvert \),
\(\left\lvert A'\cap B'\right\rvert \geq (1-\epsilon )\alpha \left\lvert B'\right\rvert \), where \(B'=B_{\rho }\) is a regular Bohr set with \({\rho \in (\tfrac {1}{2},1)\cdot c\delta \alpha /d}\), and
\[ \lVert \mu _{A'}\circ \mu _{A'}\rVert _{p(\mu _{k\cdot B''}\circ \mu _{k\cdot B''}\ast \mu _{k\cdot B'''}\circ \mu _{k\cdot B'''})} {\lt}(1+ \epsilon )\mu (B)^{-1}, \]

for any regular Bohr sets \(B'' = B'_{\rho '}\) and \(B'''=B''_{\rho ''}\) satisfying \({\rho ’,\rho ”\in (\frac{1}{2},1)\cdot c\delta \alpha /d}\).

Proof ▶

To do.

Theorem 7.7

There is a constant \(c{\gt}0\) such that the following holds. Let \(\delta ,\epsilon \in (0,1)\), let \(p \geq 1\) and let \(k\) be a positive integer such that \((k,\left\lvert G\right\rvert )=1\). There is a constant \(C=C(\epsilon ,\delta ,k){\gt}0\) such that the following holds.

For any finite abelian group \(G\) and any subset \(A\subseteq G\) with \(\left\lvert A\right\rvert =\alpha \left\lvert G\right\rvert \) there exists a regular Bohr set \(B\) with

\[ \operatorname{rank}(B)\leq Cp^4\log (2/\alpha )^5 \]

and

\[ \left\lvert B\right\rvert \geq \exp \left(-Cp^5\log (2p/\alpha )\log (2/\alpha )^6\right)\left\lvert G\right\rvert \]

and \(A' \subseteq (A-x)\cap B\) for some \(x\in G\) such that

\(\left\lvert A'\right\rvert \geq (1-\epsilon )\alpha \left\lvert B\right\rvert \),
\(\left\lvert A'\cap B'\right\rvert \geq (1-\epsilon )\alpha \left\lvert B'\right\rvert \), where \(B'=B_{\rho }\) is a regular Bohr set with \(\rho \in (\tfrac {1}{2},1)\cdot c\delta \alpha /dk\), and
\[ \lVert (\mu _{A'}-\mu _B)\ast (\mu _{A'}- \mu _B)\rVert _{p(\mu _{k\cdot B'})} \leq \epsilon \frac{\left\lvert G\right\rvert }{\left\lvert B\right\rvert }. \]

Proof ▶

To do.

Theorem 7.8

If \(A\subseteq \{ 1,\ldots ,N\} \) has size \(\left\lvert A\right\rvert =\alpha N\), then \(A\) contains at least

\[ \exp (-O(\mathcal{L}{\alpha }^{12}))N^2 \]

many three-term arithmetic progressions.

Proof ▶

To do.

Theorem 7.9 Integer case

If \(A\subseteq \{ 1,\ldots ,N\} \) contains no non-trivial three-term arithmetic progressions then

\[ \lvert A\rvert \leq \frac{N}{\exp (-c(\log N)^{1/12})} \]

Proof ▶

To do.