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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...

We prove the “Möbius and Nilsequences Conjecture” for nilsystems of step 1 and 2. This paper forms a part of our program to generalise the Hardy-Littlewood method so as to handle of linear equations in primes.

We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper [BGGT].