We present a system of interrelated conjectures which can be considered as restricted addition counterparts of classical theorems due to Kneser, Kemperman, and Scherk. Connections with the theorem of Cauchy-Davenport, conjecture of Erdős-Heilbronn, and polynomial method of Alon-Nathanson-Ruzsa are discussed.The paper assumes no expertise from the reader and can serve as an introduction to the subject.

The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an $L^2$–$L^p$ restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime $k$-tuples. Let $a_1,\cdots ,a_k$ and $b_1,\cdots ,b_k$ be positive integers. Write $h\left(\theta \right):={\sum }_{n\in X}e\left(n\theta \right)$, where $X$ is the set of all $n\le N$ such that the numbers $a_{1}n+b_1,\cdots ,a_{k}n+b_k$ are all prime. We obtain upper bounds for ${\parallel h\parallel }_{L^p\left(𝕋\right)}$, $p\>2$, which are (conditionally on the Hardy-Littlewood prime tuple conjecture) of the correct order...