By Isaac Asimov

Seventeen essays on numbers and quantity concept and the connection of numbers to dimension, the calendar, biology, astronomy, and the earth.

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The subset N(Q) = M(Q) ∩ [0, 1] is the familiar set of major arcs in the classical circle method. We also deﬁne here the minor arcs m(Q) = R\M(Q), n(Q) = [0, 1]\N(Q), although these will not be needed until the next section. 4. 4 apply also to indeﬁnite problems. With λ1 , λ2 ∈ R\{0} ﬁxed, we deﬁne K(Q1 , Q2 ) = {α ∈ R : λj α ∈ M(Qj ) (j = 1, 2)}. In addition, when y > 0, we put Ky (Q1 , Q2 ) = {α ∈ K(Q1 , Q2 ) : y < |α| ≤ 2y}. 4. Let λ1 , λ2 be non-zero real numbers such that λ1 /λ2 is irrational.

Here and later we refer to our papers “Additive representation in thin sequences” by their numeral within the series, I–VII. Hence, III refers to [9], for example. October 6, 2009 24 13:49 WSPC - Proceedings Trim Size: 9in x 6in ws-procs9x6 ¨ ¨ JORG BRUDERN, KOICHI KAWADA AND TREVOR D. WOOLEY One might object that although it is rather natural to average over the values of an integral polynomial in the case of diophantine equations, this is not adequate for inequalities, and one should take the values of a real polynomial as test points, or even a monotone sequence with a certain rate of growth.

With λ1 , λ2 ∈ R\{0} ﬁxed, we deﬁne K(Q1 , Q2 ) = {α ∈ R : λj α ∈ M(Qj ) (j = 1, 2)}. In addition, when y > 0, we put Ky (Q1 , Q2 ) = {α ∈ K(Q1 , Q2 ) : y < |α| ≤ 2y}. 4. Let λ1 , λ2 be non-zero real numbers such that λ1 /λ2 is irrational. There exists a positive real number √ ε0 = ε0 (λ1 , λ2 ) with the fol1 lowing property. Suppose that 1 ≤ Qj ≤ 2 N (j = 1, 2), and r ∈ N satisﬁes r ≤ ε0 N/(Q1 Q2 ) and rλ1 /λ2 < 1/r. Then, for any y > 0 with yN −1 Q1 Q2 r−1 . |λj |y ≥ 2Qj /N (j = 1, 2), one has mes Ky (Q1 , Q2 ) Note that N(Q) has measure about Q2 N −1 .