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Theorem 5.1

If $A_1,A_2,S\subseteq \mathbb {F}_q^n$ are such that $A_1$ and $A_2$ both have density at least $\alpha $ then there is a subspace $V$ of codimension

\[ \mathrm{codim}(V)\leq 2^{27}\mathcal{L}{(}\alpha )^2\mathcal{L}{(}\epsilon \alpha )^2\epsilon ^{-2} \]

such that

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

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Lemma 1.4

Let $A\subseteq G$ and $f:G\to \mathbb {C}$. Let $\epsilon {\gt}0$ and $m\geq 1$ and $k\geq 1$. Let

\[ L=\{ {\vec{a}}\in A^k : \| \tfrac {1}{k}\sum _{i=1}^kf(x-a_i)-\mu _A\ast f\| _{2m}\leq \epsilon \| f\| _{2m}\} . \]

If $t\in G$ is such that ${\vec{a}}\in L$ and ${\vec{a}}+(t,\ldots ,t)\in L$ then

\[ \| \tau _t(\mu _A\ast f)-\mu _A\ast f\| _{2m}\leq 2\epsilon \| f\| _{2m}. \]

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Lemma 5.3

For any function $f$ with $\sum f(x)=1$

\[ f\ast f-1/N = (f-1/N)\ast (f-1/N). \]

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Lemma 5.2

For any function $f:G\to \mathbb {C}$ and integer $k\geq 1$

\[ \| f\ast f\| _{2k}\leq \| f\circ f\| _{2k}. \]

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Lemma 6.6

There is a constant $c{\gt}0$ such that the following holds. Let $B$ be a regular Bohr set of rank $d$ and $L\geq 1$ be any integer. If $\nu :G\to \mathbb {R}_{\geq 0}$ is supported on $L B_\rho $, where $\rho \leq c/Ld$, and $\lVert \nu \rVert _1=1$, then

\[ \mu _B \leq 2\mu _{B_{1+L\rho }}\ast \nu . \]

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Definition 6.3Regularity

We say $\nu :\widehat{G}\to \mathbb {R}$ is regular if, with $d=\mathrm{rk}(\nu )$, for all $\kappa \in \mathbb {R}$ with $\left\lvert \kappa \right\rvert \leq 1/100d$ we have

\[ (1-100 d\left\lvert \kappa \right\rvert )\leq \frac{\left\lvert \mathrm{Bohr}((1+\kappa )\nu )\right\rvert }{\left\lvert \mathrm{Bohr}(\nu )\right\rvert }\leq (1+ 100 d\left\lvert \kappa \right\rvert ) \]

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Lemma 6.4

For any $\nu :\widehat{G}\to \mathbb {R}$ there exists $\rho \in [\frac{1}{2},1]$ such that $\rho \cdot \nu $ is regular.

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Definition 6.1Bohr sets

Let $\nu : \widehat{G} \to \mathbb {R}$. The corresponding Bohr set is defined to be

\[ \mathrm{Bohr}(\nu )=\left\{ x\in G : \left\lvert 1-\gamma (x)\right\rvert \leq \nu (\gamma )\textrm{ for all }\gamma \in \Gamma \right\} . \]

The rank of $\nu $, denoted by $\mathrm{rk}(\nu )$, is defined to be the size of the set of those $\gamma \in \widehat{G}$ such that $\nu (\gamma ){\lt}2$.

(Basic API facts: Bohr sets are symmetric and contain $0$. Also that, without loss of generality, we can assume $\nu $ takes only values in $\mathbb {R}_{\geq 0}$ - I think it might be easier to have the definition allow arbitrary real values, and then switch to non-negative only in proofs where convenient. Or could have the definition only allow non-negative valued functions in the first place.)

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Lemma 6.2

If $\rho \in (0,1)$ and $\nu :\widehat{G}\to \mathbb {R}$ then

\[ \left\lvert \mathrm{Bohr}(\rho \cdot \nu )\right\rvert \geq (\rho /4)^{\mathrm{rk}(\nu )}\left\lvert \mathrm{Bohr}(\nu )\right\rvert . \]

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Lemma 6.7

There is a constant $c{\gt}0$ such that the following holds. Let $B$ be a regular Bohr set of rank $d$, suppose $A\subseteq B$ has density $\alpha $, let $\epsilon {\gt}0$, and suppose $B',B''\subseteq B_\rho $ where $\rho \leq c\alpha \epsilon /d$. Then either

there is some translate $A'$ of $A$ such that $\left\lvert A'\cap B'\right\rvert \geq (1-\epsilon )\alpha \left\lvert B'\right\rvert $ and $\left\lvert A'\cap B''\right\rvert \geq (1-\epsilon )\alpha \left\lvert B''\right\rvert $, or
$\lVert 1_{A}\ast \mu _{B'}\rVert _\infty \geq (1+\epsilon /2)\alpha $, or
$\lVert 1_{A}\ast \mu _{B''}\rVert _\infty \geq (1+\epsilon /2)\alpha $.

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Theorem 2.12Chang’s lemma

Let $G$ be a finite abelian group and $f:G\to \mathbb {C}$. Let $\eta {\gt}0$ and $\alpha =N^{-1}\lVert f\rVert _1^2/\lVert f\rVert _2^2$. There exists some $\Delta \subseteq \Delta _\eta (f)$ such that

\[ \left\lvert \Delta \right\rvert \le \lceil e\mathcal{L}(\alpha )\eta ^{-2}\rceil \]

and

\[ \Delta _\eta (f)\subseteq \left\{ \sum _{\gamma \in \Delta }c_\gamma \gamma : c_\gamma \in \{ -1,0,1\} \right\} . \]

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Lemma 5.4

For any function $f$ with $\sum f(x)=1$

\[ f\circ f-1/N = (f-1/N)\circ (f-1/N). \]

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Proposition 5.6

Let $\epsilon \in (0,1)$. If $A,C\subseteq \mathbb {F}_q^n$, where $C$ has density at least $\gamma $, and

\[ \left\lvert N\langle \mu _A\ast \mu _A,\mu _C\rangle -1\right\rvert {\gt} \epsilon \]

then there is a subspace $V$ of codimension

\[ \leq 2^{171}\epsilon ^{-24}\mathcal{L}{(}\alpha )^4\mathcal{L}{(}\gamma )^4. \]

such that $\lVert 1_{A}\ast \mu _V\rVert _\infty \geq (1+\epsilon /32)\alpha $.

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Lemma 2.10

If $A\subseteq G$ is dissociated then $E_{2m}(A) \le (32e m \left\lvert A\right\rvert )^m$.

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Lemma 2.11

If $A\subseteq G$ contains no dissociated set with $\geq K+1$ elements then there is $A'\subseteq A$ of size $\left\lvert A'\right\rvert \le K$ such that

\[ A\subseteq \left\{ \sum _{a\in A'}c_aa : c_a\in \{ -1,0,1\} \right\} . \]

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Definition 2.1Dissociation

We say that $A\subseteq G$ is dissociated if, for any $m\geq 1$, and any $x\in G$, there is at most one $A'\subset A$ of size $\left\lvert A'\right\rvert =m$ such that

\[ \sum _{a\in A'}a=x. \]

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Lemma 4.1

Let $p\geq 2$ be an even integer. Let $B_1,B_2\subseteq G$ and $\mu =\mu _{B_1}\circ \mu _{B_2}$. For any finite set $A\subseteq G$ and function $f:G\to \mathbb {R}_{\geq 0}$ there exist $A_1\subseteq B_1$ and $A_2\subseteq B_2$ such that

\[ \langle \mu _{A_1}\circ \mu _{A_2}, f\rangle \lVert 1_{A}\circ 1_{A}\rVert _{p(\mu )}^p\leq 2\langle (1_{A}\circ 1_{A})^p,f\rangle _\mu \]

and

\[ \min \left( \frac{\left\lvert A_1\right\rvert }{\left\lvert B_1\right\rvert },\frac{\left\lvert A_2\right\rvert }{\left\lvert B_2\right\rvert }\right)\geq \frac{1}{4}\left\lvert A\right\rvert ^{-2p}\lVert 1_{A}\circ 1_{A}\rVert _{p(\mu )}^{2p}. \]

LaTeX Lean

Definition 2.6Energy

Let $G$ be a finite abelian group and $A\subseteq G$. Let $m\geq 1$. We define

\[ E_{2m}(A)=\sum _{a_1,\ldots ,a_{2m}\in A}1_{a_1+\cdots -a_{2m}=0}. \]

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Lemma 2.9

If $A\subset G$ and $m\geq 1$ then

\[ E_{2m}(A) = \sum _x 1_A^{(m)}(x)^2. \]

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Theorem 5.8

Let $q$ be an odd prime power. If $A\subseteq \mathbb {F}_q^n$ with $\alpha =\left\lvert A\right\rvert /q^n$ has no non-trivial three-term arithmetic progressions then

\[ n \ll \mathcal{L}{(}\alpha )^9. \]

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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 ${\rho ’,\rho ”\in (\frac{1}{2},1)\cdot c\delta \alpha /d}$.

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Lemma 2.7

Let $G$ be a finite abelian group and $f:G\to \mathbb {C}$. Let $\nu :G\to \mathbb {R}_{\geq 0}$ be such that whenever $\left\lvert f\right\rvert \neq 0$ we have $\nu \geq 1$. Let $\Delta \subseteq \Delta _\eta (f)$. Then, for any $m\geq 1$.

\[ \eta ^{2m}\frac{\lVert f\rVert _1^2}{\lVert f\rVert _2^2}\left\lvert \Delta \right\rvert ^{2m}\le E_{2m}(\Delta ;\nu ). \]

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Lemma 5.5

Let $\epsilon {\gt}0$ and $\mu \equiv 1/N$. If $A,C\subseteq G$, where $C$ has density at least $\gamma $, and

\[ \left\lvert N\langle \mu _A\ast \mu _A,\mu _C\rangle -1\right\rvert {\gt}\epsilon \]

then, if $f=(\mu _A-1/N)$, $\lVert f\circ f\rVert _{p(\mu )} \geq \epsilon /2N$ for $p=2\lceil \mathcal{L}{(}\gamma )\rceil $.

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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 $\lVert (\mu _A-\mu _B)*(\mu _A-\mu _B)\rVert _{p(\mu _{B'})} \geq \tfrac {1}{2}\epsilon \mu (B)^{-1}$.

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Theorem 7.9Integer case

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

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

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Definition 2.4Large spectrum

Let $G$ be a finite abelian group and $f:G\to \mathbb {C}$. Let $\eta \in \mathbb {R}$. The $\eta $-large spectrum is defined to be

\[ \Delta _\eta (f) = \{ \gamma \in \widehat{G} : \lvert \widehat{f}(\gamma )\rvert \geq \eta \lVert f\rVert _1\} . \]

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Theorem 1.7$L_\infty $ almost periodicity

Let $\epsilon \in (0,1]$. Let $K\geq 2$ and $A,S\subseteq G$ with $\lvert A+S\rvert \leq K\lvert A\rvert $. Let $B,C\subseteq G$. Let $\eta =\min (1,\lvert C\rvert /\lvert B\rvert )$. There exists $T\subseteq G$ such that

\[ \lvert T\rvert \geq K^{-4096\lceil \mathcal{L}{\eta }\rceil \epsilon ^{-2}}\lvert S\rvert \]

such that for any $t\in T$ we have

\[ \| \tau _t(\mu _A\ast 1_B\ast \mu _C)-\mu _A\ast 1_B\ast \mu _C\| _{\infty }\leq \epsilon . \]

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Theorem 1.8

Let $\epsilon \in (0,1]$ and $k\geq 1$. Let $K\geq 2$ and $A,S\subseteq G$ with $\lvert A+S\rvert \leq K\lvert A\rvert $. Let $B,C\subseteq G$. Let $\eta =\min (1,\lvert C\rvert /\lvert B\rvert )$. There exists $T\subseteq G$ such that

\[ \lvert T\rvert \geq K^{-4096\lceil \mathcal{L}{\eta }\rceil k^2\epsilon ^{-2}}\lvert S\rvert \]

such that

\[ \| \mu _T^{(k)}\ast \mu _A\ast 1_B\ast \mu _C-\mu _A\ast 1_B\ast \mu _C\| _{\infty }\leq \epsilon . \]

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Lemma 1.5

Let $A\subseteq G$ and $k\geq 1$ and $L\subseteq A^k$. Then there exists some ${\vec{a}}\in L$ such that

\[ \# \{ t\in G : {\vec{a}}+(t,\ldots ,t)\in L\} \geq \frac{\lvert L\rvert }{\lvert A+S\rvert ^k}\lvert S\rvert . \]

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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

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

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Theorem 1.6$L_p$ almost periodicity

Let $\epsilon \in (0,1]$ and $m\geq 1$. Let $K\geq 2$ and $A,S\subseteq G$ with $\lvert A+S\rvert \leq K\lvert A\rvert $. Let $f:G\to \mathbb {C}$. There exists $T\subseteq G$ such that

\[ \lvert T\rvert \geq K^{-512m\epsilon ^{-2}}\lvert S\rvert \]

such that for any $t\in T$ we have

\[ \| \tau _t(\mu _A\ast f)-\mu _A\ast f\| _{2m}\leq \epsilon \| f\| _{2m}. \]

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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.

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Lemma 1.1Marcinkiewicz-Zygmund inequality

Let $m\geq 1$. If $f:G\to \mathbb {R}$ is such that $\mathbb {E}_x f(x)=0$ and $\left\lvert f(x)\right\rvert \leq 2$ for all $x$ then

\[ \mathbb {E}_{x_1,\ldots ,x_n} \left\lvert \sum _{i=1}^n f(x_i)\right\rvert ^{2m} \leq (4mn)^{m}. \]

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Lean

Real.marcinkiewicz_zygmund'
Real.marcinkiewicz_zygmund

Lemma 1.2Complex-valued Marcinkiewicz-Zygmund inequality

Let $m\geq 1$. If $f:G\to \mathbb {C}$ is such that $\mathbb {E}_x f(x)=0$ and $\left\lvert f(x)\right\rvert \leq 2$ for all $x$ then

\[ \mathbb {E}_{x_1,\ldots ,x_n} \left\lvert \sum _{i=1}^n f(x_i)\right\rvert ^{2m} \leq (8mn)^{m}. \]

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Lemma 5.7

If $A\subseteq G$ has no non-trivial three-term arithmetic progressions and $G$ has odd order then

\[ \langle \mu _A\ast \mu _A,\mu _{2\cdot A}\rangle = 1/\left\lvert A\right\rvert ^2. \]

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Lemma 3.1

For any function $f:G\to \mathbb {R}$ and integer $k\geq 0$

\[ \mathbb {E}_x f\circ f(x)^k\geq 0. \]

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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

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

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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

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

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Lemma 1.3

Let $\epsilon {\gt}0$ and $m\geq 1$. Let $A\subseteq G$ and $f:G\to \mathbb {C}$. If $k\geq 64m\epsilon ^{-2}$ then the set

\[ L=\{ {\vec{a}}\in A^k : \| \tfrac {1}{k}\sum _{i=1}^kf(x-a_i)-\mu _A\ast f\| _{2m}\leq \epsilon \| f\| _{2m}\} . \]

has size at least $\lvert A \rvert ^k/2$.

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Lemma 6.5

If $B$ is a regular Bohr set of rank $d$ and $\mu :G\to \mathbb {R}_{\geq 0}$ is supported on $B_\rho $, with $\rho \in (0,1)$, then

\[ \lVert \mu _B*\mu - \mu _B \rVert _{1} \ll \rho d\lVert \mu \rVert _1. \]

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Lemma 2.3Rudin’s inequality

If the discrete Fourier transform of $f : G \longrightarrow \mathbb {C}$ has dissociated support and $p \ge 2$ is an integer, then $\lVert f\rVert _p \le 4\sqrt{pe} \lVert f\rVert _2$.

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Lemma 2.2Rudin’s exponential inequality

If the discrete Fourier transform of $f : G \longrightarrow \mathbb {C}$ has dissociated support, then

\[ \mathop{ \mathchoice {\vcenter {\hbox{{\@tempdima +4\p@ \@tempdima 2.0002\@tempdima \divide \@tempdima \p@ \@tempcnta \@tempdima \@tempdimb \f@size \p@ \def\@tempa {1.09545}\@whilenum {\@tempcnta {\gt}\z@ }\do {\advance \m@ne \@tempdimb 1.09545\@tempdimb }\PackageWarning{relsize}{Font size ??? is too small.\MessageBreak Using 999pt instead}\@tempdimb =999pt\def\@tempa {\relax }\@tempdima -\@m \p@ \let\rs@size \rs@lookup \@tempdima 100\@tempdima }\relax $\mathbb {E}$}}} {\kern 0pt\mathbb {E}} {\kern 0pt\mathbb {E}} {\kern 0pt\mathbb {E}} }\displaylimits \exp (\Re f) \le \exp \left(\frac{\lVert f\rVert _2^2}2\right) \]

It follows that

\[ \mathop{ \mathchoice {\vcenter {\hbox{{\@tempdima +4\p@ \@tempdima 2.0002\@tempdima \divide \@tempdima \p@ \@tempcnta \@tempdima \@tempdimb \f@size \p@ \def\@tempa {1.09545}\@whilenum {\@tempcnta {\gt}\z@ }\do {\advance \m@ne \@tempdimb 1.09545\@tempdimb }\PackageWarning{relsize}{Font size ??? is too small.\MessageBreak Using 999pt instead}\@tempdimb =999pt\def\@tempa {\relax }\@tempdima -\@m \p@ \let\rs@size \rs@lookup \@tempdima 100\@tempdima }\relax $\mathbb {E}$}}} {\kern 0pt\mathbb {E}} {\kern 0pt\mathbb {E}} {\kern 0pt\mathbb {E}} }\displaylimits _x e^{\left\lvert f(x)\right\rvert } \le 2e^{\| f\| _2^2/2}. \]

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Lemma 4.2

Let $\epsilon ,\delta {\gt}0$ and $p\geq \max (2,\epsilon ^{-1}\log (2/\delta ))$ be an even integer. Let $B_1,B_2\subseteq G$, and let $\mu =\mu _{B_1}\circ \mu _{B_2}$. For any finite set $A\subseteq G$, if

\[ S=\{ x\in G: 1_{A}\circ 1_{A}(x){\gt}(1-\epsilon )\lVert 1_{A}\circ 1_{A}\rVert _{p(\mu )}\} , \]

then there are $A_1\subseteq B_1$ and $A_2\subseteq B_2$ such that

\[ \langle \mu _{A_1}\circ \mu _{A_2},1_{S}\rangle \geq 1-\delta \]

and

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Corollary 4.3

Let $\epsilon ,\delta {\gt}0$ and $p\geq \max (2,\epsilon ^{-1}\log (2/\delta ))$ be an even integer and $\mu \equiv 1/N$. If $A\subseteq G$ has density $\alpha $ and

\[ S = \{ x : \mu _A\circ \mu _A(x) \geq (1-\epsilon )\| \mu _A\circ \mu _A\| _{p(\mu )}\} \]

then there are $A_1,A_2\subseteq G$ such that

\[ \langle \mu _{A_1}\circ \mu _{A_2}, 1_S\rangle \geq 1-\delta \]

and both $A_1$ and $A_2$ have density

\[ \geq \frac{1}{4}\alpha ^{2p}. \]

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Lemma 2.8

Let $G$ be a finite abelian group and $f:G\to \mathbb {C}$. Let $\Delta \subseteq \Delta _\eta (f)$. Then, for any $m\geq 1$.

\[ N^{-1}\eta ^{2m}\frac{\lVert f\rVert _1^2}{\lVert f\rVert _2^2}\left\lvert \Delta \right\rvert ^{2m}\le E_{2m}(\Delta ). \]

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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

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

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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
\[ \lVert (\mu _{A'}-\mu _B)\ast (\mu _{A'}- \mu _B)\rVert _{p(\mu _{k\cdot B'})} \leq \epsilon \frac{\left\lvert G\right\rvert }{\left\lvert B\right\rvert }. \]

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Lemma 3.2

Let $\epsilon \in (0,1)$ and $\nu :G\to \mathbb {R}_{\geq 0}$ be some probability measure such that $\widehat{\nu }\geq 0$. Let $f:G\to \mathbb {R}$. If $\lVert f\circ f\rVert _{p(\nu )}\geq \epsilon $ for some $p\geq 1$ then $\lVert f\circ f+1\rVert _{p'(\nu )}\geq 1+\tfrac {1}{2}\epsilon $ for $p'=120\epsilon ^{-1}\log (3/\epsilon )$.

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Definition 2.5Weighted energy

Let $\Delta \subseteq \widehat{G}$ and $m\geq 1$. Let $\nu :G\to \mathbb {C}$. Then

\[ E_{2m}(\Delta ;\nu )=\sum _{\gamma _1,\ldots ,\gamma _{2m}\in \Delta }\left\lvert \widehat{\nu }(\gamma _1+\cdots -\gamma _{2m})\right\rvert . \]

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