Dependencies

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

such that

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

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

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

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

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

and

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

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

then there is a subspace \(V\) of codimension

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

If \(A\subseteq G\) is dissociated then \(E_{2m}(A) \leq (8e m \left\lvert A\right\rvert )^m\).

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

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

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

and

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

If \(A\subset G\) and \(m\geq 1\) then

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

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 \(m\geq 1\).

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

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

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

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 \(B,C\subseteq G\). Let \(\eta =\min (1,\lvert C\rvert /\lvert B\rvert )\). There exists \(T\subseteq G\) such that

such that for any \(t\in T\) we have

Let \(\epsilon \in (0,1]\) and \(m\geq 1\) 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

such that

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

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

such that for any \(t\in T\) we have

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

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

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

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

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

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

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

If the discrete Fourier transform of \(f : G \longrightarrow \mathbb {C}\) has dissociated support, then \(\mathbb {E}\exp (\Re f) \le \exp (\frac{\lVert f\rVert _2^2}2)\).

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

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

and

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

then there are \(A_1,A_2\subseteq G\) such that

and both \(A_1\) and \(A_2\) have density

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

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 \(p'=120\epsilon ^{-1}\log (3/\epsilon )\).

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