By Melvyn B. Nathanson (auth.), David Chudnovsky, Gregory Chudnovsky (eds.)
This amazing quantity is devoted to Mel Nathanson, a number one authoritative specialist for numerous a long time within the zone of combinatorial and additive quantity conception. Nathanson's various effects were greatly released in first-class journals and in a couple of first-class graduate textbooks (GTM Springer) and reference works. For a number of a long time, Mel Nathanson's seminal principles and leads to combinatorial and additive quantity conception have encouraged graduate scholars and researchers alike. The invited survey articles during this quantity replicate the paintings of exotic mathematicians in quantity idea, and signify a variety of very important themes in present research.
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Additional info for Additive Number Theory: Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson
O’Bryant Problem 2. Find a necessary and sufficient condition on the real numbers ˛i ; ˇj 2R such that for all positive integers n, b bbn˛1 c ˛2 c ˛d c D b bbnˇ1 c ˇ2 c ˇ` c: There are very many solutions in rationals, and we do not have a guess as to their structure. Both problems are obvious if all ˛; ˇ are taken to be integers, and both are answered here if d D ` and the ˛; ˇ are taken to be sufficiently irrational. The most difficult case to understand, for both questions, seems to be when the ˛; ˇ are all rational, but not all integral.
Let H < Fp ; jH j D p ˛ for some ˛ > 0. t u 18 J. Bourgain on Fp is H -invariant provided Definition. hk/ for all k 2 Fp ; h 2 H: Example. x/ D 1 jH j if x 2 H : 0 if x 62 H: Main Proposition. ˛; ; ı/ > 0 such that ƒı 0 6D f0g ) jƒı j > p : Here, ƒı D ƒı . /, where is an arbitrary H -invariant measure. ˛; ; ı/ D . 1 C exp. 1 ˛ // The limitation of the method: jH j D p with ˛ 1 log log p (see [B1]). Proof of Theorem using Proposition. Take D 1 ˛3 ; ı D ˛4 ) ı 0 , according to the Proposition. Apply the Proposition with D jH1 j 1H .
Q; u/ D 1. , the reason we need ˛ to be nonintegral. q; u/ D 1: u t Proof (Proof of Theorem 2). We proceed by induction on d . The claim is immediate for d D 1. Now assume that d 2 and that Theorem 2 holds for d 1. Assume without loss of generality that ˛1 ˛2 ˛d . If bn˛1 c D q is prime, Q then it will show up in the factorization of diD1 bn˛i c D Pn as a prime factor 1=d q Pn (since bn˛1 c bn˛i c for all i ). Conversely, any prime factor q of Pn 1=d which is greater than or equal to Pn must come from bn˛1 c.
Additive Number Theory: Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson by Melvyn B. Nathanson (auth.), David Chudnovsky, Gregory Chudnovsky (eds.)