Arithmetical Aspects of the Large Sieve Inequality

The large sieve inequality for integer polynomial amplitudes

“…A number of authors (see [1,7,8]) have recently obtained upper bounds for the sum on the lefthand side of (2) from various points of view. These bounds are, however, comparable to that given by when P (T ) = T k , for any integer k 3, and when I is of the form (0, N] then the left-hand side of (2) is (N Q 2(1−1/k) + Q 2 )N 1+ sup 0<i N |a i | 2 under Hooley's hypothesis K * in the context of Waring's problem.…”

“…When indeed P (T ) is of degree 2 and the interval I is of the form (0, N], Ramaré's method, described in Section 5.4 of [7], gives the bound Q (N + Q g(Q ))(log 2 2Q ) 2 for the sum on the lefthand side of (2), where g(Q ) = exp(C log 2 Q log 3 Q ). Here log 2 Q and log 3 Q denote log log Q and log log log Q , respectively.…”

On long κ-tuples with few prime factors

“…is prime to d, for all i from 1 to κ. We proposed some time ago (see [20] and [19]) a geometrical approach that dispenses with building an auxiliary polynomial, like Π (h1,••• ,hκ) above. The exposition will be made easier by the following definition.…”

“…Let M = lcm(d ≤ Q) and let us look at K M ⊂ Z/M Z. We assume momentarily that the compact set satisfies the Johnsen-Gallagher condition (see [6], [23], [20], [19]…”

Gowers norms of multiplicative functions in progressions on average

“…This is the reason that we are unable to prove Theorem 1.2 for the primes, which would require type-II estimates in an inaccessible range, and also the reason that one saves no more than log X in the bound (1.1). In fact, to get this saving we use a quantitatively superior argument of Ramaré [16], which received a lot of attention recently [13,4] following its use in Matomäki and Radziwi l l's recent breakthrough [12]. Finally the required type-II estimates will be proved in Section 4.…”