By Melvyn B. Nathanson (auth.), David Chudnovsky, Gregory Chudnovsky (eds.)

ISBN-10: 0387370293

ISBN-13: 9780387370293

ISBN-10: 0387683615

ISBN-13: 9780387683614

This extraordinary quantity is devoted to Mel Nathanson, a number one authoritative professional for numerous many years within the region of combinatorial and additive quantity conception. Nathanson's various effects were broadly released in top quality 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 rules and leads to combinatorial and additive quantity thought have inspired graduate scholars and researchers alike. The invited survey articles during this quantity mirror the paintings of individual mathematicians in quantity conception, and characterize quite a lot of very important themes in present research.

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Extra info for Additive Number Theory: Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson

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Sketch of the Proof. d C 1/m1 . mod p m1 /. Hence gs1 D 1 C bp m1 . 1 C bp m1 k / Á X Âk à j Äd This is a polynomial in k of degree d < j m m1 . mod p m /: Sum-Product Theorems and Applications 31 1 1 Take m1 with p m1 < t 4 and N t 4 . Write X X epm . x; y/ D X Âxy à j Äd j bj 1 pm j m1 : Apply then Vinogradov exponential sum bound. log t/3 : There is the following general multilinear bound for composite modulus. Theorem 5 ([B3]). 1 Ä i Ä k/ and also max jAi \ 2Zq1 Then 1 q1 . x1 : : : xk /ˇˇ < q ˇ ˇ ˇx1 2A1 ;:::;xk 2Ak ı jA1 j jAk j: 10 Exponential Sums in Finite Commutative Rings Let R be a finite commutative ring with unit and assume jRj D q with no small prime divisors: Denote R D invertible elements.

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. In addition to its usual meaning, let “<” denote the lexicographic ordering on f0; 1gd . , the number of 1s in W . Lemma 3. V; ˛I N 1/: Proof. We work by induction on d . For d D 1, the result obviously holds since ˛1 62 Z. 50 R. Graham and K. d 1/-tuples . Assume that W < V . v2 vd ; hŒ˛1 w1 ˛2 ; ˛3 ; : : : ; ˛d iI 1/ Thus, we may assume that w1 < v1 , and so w1 wd 1 < v1 vd 1 .

R2 ) the number of real (resp. I // for almost all ideals I (Egami’s problem). (results by Konyagin–Shparlinski, Bourgain–Chang, . . ). More precise statements obtained in [B-C2]. Prime ideal case. T / the number of prime ideals of norm ÄT . 1// T : log T Theorem 1 ([B-C2]). "; K/ > 0 such that for T ! P/1 ı gj < T " : General integral ideals. I / Ä T . K/j: Theorem 2 ([B-C2]). ı/ ! 0 with ı ! I /1 ı for ideals I outside a sequence of asymptotic density at most ı 0 . 34 J. Bourgain Main idea. Consider the quotient map ' W O !

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Additive Number Theory: Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson by Melvyn B. Nathanson (auth.), David Chudnovsky, Gregory Chudnovsky (eds.)

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