By Melvyn B. Nathanson (auth.), David Chudnovsky, Gregory Chudnovsky (eds.)
This extraordinary quantity is devoted to Mel Nathanson, a number one authoritative professional for numerous a long time within the region of combinatorial and additive quantity concept. Nathanson's quite a few effects were extensively released in first-class journals and in a couple of very good 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 idea have stimulated graduate scholars and researchers alike. The invited survey articles during this quantity mirror the paintings of individual mathematicians in quantity concept, and characterize quite a lot of very important subject matters in present research.
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Additional info for Additive Number Theory: Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson
1 N nD1 T3;2 D lim Z 1 Z 1 Z 1 x˛2 ˛3 C fy 0 0 bbbn˛1 c ˛2 c˛3 c/ x˛2 g˛3 C fz x˛2 ˛3 fy x˛2 g˛3 gdx dy dz: 0 Using ˛N 2 Œ0; 1/3 , we can eliminate the fractional parts in the above integral and get T3;2 D 2 C 3˛2 C 2˛22 2 1 C ˛2 1 C ˛3 C ˛3 : 3 2 6 It is clear that this method can yield a formula for Td;k for any d; k. 5 Proof of Theorem 4 Let Œx0 be the floor of x, and Œx1 be the ceiling. W; ˛I N n/ WD Œ: : : ŒŒn˛1 w1 ˛2 w2 : : : ˛d wd ; where W D w1 w2 : : : wk is a word in the alphabet f0; 1g, and ˛N D h˛1 ; ˛2 ; : : : ; ˛d i.
Use the circle method. A/ ˇ < max ˇ ˇ ˇ p ˇ a2Fp ˇ 2k" jAj2k : x2A t u Conversely, we have the following: Lemma 2. H / < p 1 2 ı jH j4k 22 J. Bourgain for some k 2 ZC and ı > 0. ax/ˇ Ä p ˇ ˇ a2Fp ı 4k 2 jH j: x2H Proof. axz/ ˇ ˇ p ˇ ˇ jH j2k 0 Ä z2Fp (H¨older) Ä D (Hadamard) Ä proving Lemma 2. ax/ˇ Ä pjH j; max ˇ ˇ ˇ a2Fp x2H p which is nontrivial for jH j > p. Prior to [B-G-K], completely explicit Gauss-sum estimates (with power-saving) 1 for smaller groups (up to jH j > p 4 C" / had been obtained using variants of Stepanov’s method (Garcia–Voloch, Shparlinski, Heath–Brown, Heath-BrownKonyagin, Konyagin).
Graham and K. O’Bryant Proof. First, observe that the sequence contains all large positive integers, if 0 < j˛j Ä 1, so we assume henceforth that j˛j > 1. First, we further assume that ˛ is irrational and positive. bn˛ C c/1 nD1 contains arbitrarily large primes. k /=˛ 2 Z. Thus it suffices for our purposes to show that the sequence of fractional parts fp=˛g is uniformly distributed, where p goes through the prime numbers. This was shown by Vinogradov [7, Chapter XI]. If ˛ is irrational and negative, then jbn˛cj D bnj˛j C 1c, and this is the case considered in the previous paragraph.