By Edmund Landau
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R. Math. Acad. Sci. Paris 342(9) (2006), 643–646. [B-C2] J. Bourgain, M. Chang, On the minimum norm of representatives of residue classes in number fields, Duke Math. J. 138(2) (2007), 263–280. [B-C3] J. Bourgain, M. Chang, Exponential sum estimates over subgroups and almost subgroups of ZQ , where Q is composite with few prime factors, GAFA 16(2) (2006), 327–366. [B-G] J. Z. Garaev, On a variant of sum-product estimates and explicit exponential sum bounds in prime fields, Math. Proc. Camb. Phil.
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 . V; ˛I N 1/: Lemma 4. V; ˛I N 1/. 0d ; ˇI then ff˛i gW 1 Ä i Ä d g D ffˇi gW 1 Ä i Ä d g. Proof. ˇ/W Can You Hear the Shape of a Beatty Sequence? ˛/ N C Az is determined by the sequence. Therefore, the set of its roots 1 f˛f˛i gi g is also determined by the sequence. Since x 7! 1 x x is a 1-1 map, this implies that the set ff˛1 g; : : : ; f˛d gg is determined from the sequence, concluding the proof.
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). 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 f x ;  y ;  xy g.
