LINEAR EQUATIONS OVER Fp AND MOMENTS OF EXPONENTIAL SUMS

Two of our principal results (in a simplified form) are as follows. theorem For p prime, the number of solutions of the equation c1a1 + · · · + ckak = λ, aj ∈ Aj , where cj ∈ F×p and λ ∈ Fp are fixed coefficients, and the variables aj range over sets Aj ⊆ Fp, does not exceed the number of solutions of the equation a1 + · · · + ak = 0, aj ∈ Āj , where the variables aj range over arithmetic progressions Āj ⊆ Fp of cardinalities |Āj | = |Aj |, balanced around zero. This readily implies an integer version, which strengthens a result of R. Gabriel, G. Hardy, and J. Littlewood. theorem Let A be a set of n = |A| residues modulo a prime p. For z ∈ Fp, write SA(z) = ∑ a∈A e2πi(az/p). Then for ε > 0 we have # { z ∈ F×p : ∣∣SA(z)∣∣ > (1− ε)n} ≤ 2 √ 6 π p n ε1/2 ( 1+ o(1)), provided n → ∞ and ε → 0. Equality is attained when A is an arithmetic progression modulo p. DUKE MATHEMATICAL JOURNAL Vol. 107, No. 2, c © 2001 Received 24 August 1999. 2000 Mathematics Subject Classification. Primary 11B75; Secondary 42C20, 11D04, 11L07, 11P99, 11B25, 11D12. Author’s work partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).