Let $\Sigma$ be an alphabet and $\mu$ be a distribution on $\Sigma ^k$ for some $k \geqslant 2$. Let $\alpha \gt 0$ be the minimum probability of a tuple in the support of $\mu$ (denoted $\mathsf{supp}(\mu )$). We treat the parameters $\Sigma , k, \mu , \alpha$ as fixed and constant. We say that the distribution $\mu$ has a linear embedding if there exist an Abelian group $G$ (with the identity element $0_G$) and mappings $\sigma _i : \Sigma \rightarrow G$, $1 \leqslant i \leqslant k$, such that at least one of the mappings is non-constant and for every $(a_1, a_2, \ldots , a_k)\in \mathsf{supp}(\mu )$, $\sum _{i=1}^k \sigma _i(a_i) = 0_G$. In [Bhangale-Khot-Minzer, STOC 2022], the authors asked the following analytical question. Let $f_i: \Sigma ^n\rightarrow [\!-1,1]$ be bounded functions, such that at least one of the functions $f_i$ essentially has degree at least $d$, meaning that the Fourier mass of $f_i$ on terms of degree less than $d$ is at most $\delta$. If $\mu$ has no linear embedding (over any Abelian group), then is it necessarily the case that

\begin{equation*}\left | \mathop {\mathbb{E}}_{({\textbf {x}}_1, {\textbf {x}}_2, \ldots , {\textbf {x}}_k)\sim \mu ^{\otimes n}}[f_1({\textbf {x}}_1)f_2({\textbf {x}}_2)\cdots f_k({\textbf {x}}_k)] \right | = o_{d, \delta }(1),\end{equation*}

$\to 0$

$d \to \infty$

$\delta \to 0$

In this paper, we answer this analytical question fully and in the affirmative for $k=3$. We also show the following two applications of the result.

1. 1. The first application is related to hardness of approximation. Using the reduction from [5], we show that for every $3$-ary predicate $P:\Sigma ^3 \to \{0,1\}$ such that $P$ has no linear embedding, an SDP (semi-definite programming) integrality gap instance of a $P$-Constraint Satisfaction Problem (CSP) instance with gap $(1,s)$ can be translated into a dictatorship test with completeness $1$ and soundness $s+o(1)$, under certain additional conditions on the instance.

2. 2. The second application is related to additive combinatorics. We show that if the distribution $\mu$ on $\Sigma ^3$ has no linear embedding, marginals of $\mu$ are uniform on $\Sigma$, and $(a,a,a)\in \texttt{supp}(\mu )$ for every $a\in \Sigma$, then every large enough subset of $\Sigma ^n$ contains a triple $({\textbf {x}}_1, {\textbf {x}}_2,{\textbf {x}}_3)$ from $\mu ^{\otimes n}$ (and in fact a significant density of such triples).