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

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

This outstanding quantity is devoted to Mel Nathanson, a number one authoritative professional for numerous a long time within the region of combinatorial and additive quantity thought. Nathanson's various effects were extensively released in first class journals and in a few very good graduate textbooks (GTM Springer) and reference works. For a number of many years, Mel Nathanson's seminal rules and ends up in combinatorial and additive quantity conception have stimulated graduate scholars and researchers alike. The invited survey articles during this quantity mirror the paintings of unusual mathematicians in quantity concept, and signify quite a lot of very important issues in present research.

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Theorem 2 ([B-C]). " > 0 arbitrary). "/ > 0. 8 The Case of General Polynomial (mod p) We first recall Weil’s estimate. Theorem 1 (Weil). x/ 2 Fp ŒX  of degree d . x// p ˇ ˇ ˇ ˇ1ÄxÄp Problem. Obtain non-trivial estimates for d p p. Sum-product technology enables one to obtain such results for special (sparse) polynomials (as considered by Mordell, cf. [Mor]). Theorem 2 ([B2]). 1 Ä i 6D j < r/: 28 J. ı > 0 arbitrary). The following example shows that the second condition is necessary. Example (Cochrane–Pinner).

Michel, A. Venkatesh, Some effective results for a; b, ETDS, vol 29, 06 (2009), 1705–1722. [Ga] M. Garaev, An explicit sum-product estimate in Fp , IMRN (2007), no 11. R. C. G. J. ), 451–463 (Progress in Mathematics 138/139, Birkh¨auser-Verlag, Boston 1996). R. V. Konyagin, New bounds for Gauss sum derived from k-th powers and for Heilbronn’s exponential sum, Q. J. Math. 51 (2003), 221–335. [Ka-S] N. -Y. Shen, A slight improvement to Garaev sum-product estimate, Proc. AMS 136(7) (2008), 2499–2504.

Mor]). Theorem 2 ([B2]). 1 Ä i 6D j < r/: 28 J. ı > 0 arbitrary). The following example shows that the second condition is necessary. Example (Cochrane–Pinner). x// D 1 2 p D C x: X ep . p /D 1 1 2 p C 0. p/: Theorem 3. r; ı/. Applications to cryptography and distributional properties of Diffie–Hellman triples fÂ x ; Â y ; Â xy g. Power generators unC1 D uen . e D 2/. Theorems 2 and 3 rely on an extension of the sum-product theorem to Cartesian products. 0 Theorem 4. Fix " > 0. ı/ ! x; ax/jx 2 Fp g such that jA \ Lj > p 1 ı0 : 9 The Sum-Product in Zq D Z=qZ Because of the presence of subrings when q is composite, additional restrictions on A Zq are needed.