The size of exponential sums on intervals of the real line

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We prove that there is a constant c > 0 depending only on M ≥ 1 and μ≥ 0 such that ∫yʸ⁺ᵃ|g(t)| dt ≥ (-c/(aδ)), a ∈ (0,π], for every g of the form g(t) = ∑ⱼ₌₀ⁿaⱼ eiλⱼt, aⱼ ∈ C, |aⱼ| ≤ Mjμ, |a₀|=1, n ∈ N, where the exponents λⱼ ∈ C satisfy Re(λ₀) = 0, Re(λⱼ) ≥ jδ> 0, j=1,2,, and for every subinterval [y,y+a] of the real line. Establishing inequalities of this variety is motivated by problems in physics.