an:03985344
Zbl 0609.10042
Ruzsa, Imre Z.
Essential components
EN
Proc. Lond. Math. Soc., III. Ser. 54, 38-56 (1987).
00151050
1987
j
11B05 11B13 11K99 11P55 11B83
sum-sets; sets of integers; asymptotic density; trigonometric sums; essential component
The author calls a set H of nonnegative integers an essential component if ???(A\(+H)>\underline d(A)\) for any set A with ???(A)\(<1\). (Here ??? denotes lower asymptotic, not Schnirelman density!). It is proved by probabilistic methods that there exist, for every \(\epsilon >0\), essential components satisfying \(H(x)=O(\log^{1+\epsilon} x)\). Furthermore, it is shown that, for any essential component H, there exist numbers \(c>0\) and \(x_ 0\) such that \(H(x)>\log^{1+\epsilon} x\), \(\forall x>x_ 0\). One of the main tools is a characterization of essential components in terms of additive behavior modulo m \((m=1,2,...)\).
B.Volkmann