In the present paper we investigate distributional properties of sparse sequences modulo almost all prime numbers. We obtain new results for a wide class of sparse sequences which in particular find applications on additive problems and the discrete Littlewood problem related to lower bound estimates of the L1-norm of trigonometric sums.

Here, we show that for most primes p, every residue class modulo p can be represented as a sum of 32 Fibonacci numbers.

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Let f (z) = ∞ n=1 a(n)e 2πinz be a normalized Hecke eigenform in S new 2k (Γ 0 (N )) with integer Fourier coefficients. We prove that there exists a constant C(f ) > 0 such that any integer is a sum of at most C(f ) coefficients a(n). It holds C(f ) ≪ ε,k N 6k−3 16 +ε .

Abstract. Let τ (n) be the Ramanujan τ -function. We prove that for any integer N with |N| ≥ 2 the diophantine equation 148000