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В работе получена асимптотическая формула для количества представлений натурального числа N в виде q1 + q2 + [αq3], где q1, q2, q3 --бесквадратные числа, α > 1 --фиксированное иррациональное алгебраическое число.Ключевые слова: тернарные задачи, бесквадратные числа, асимптотическая формула.

Homogeneous Beatty sequences are sequences of the form 𝑎 𝑛 = [𝛼𝑛], where 𝛼 is a positive irrational number. In 1957 T. Skolem showed that if the numbers 1, 1 𝛼 , 1 𝛽 are linearly independent over the field of rational numbers, then the sequences [𝛼𝑛] and [𝛽𝑛] have infinitely many elements in common. T. Bang strengthened this result: denote 𝑆 𝛼,𝛽 (𝑁 ) the number of natural numbers 𝑘, 1 𝑘 𝑁 , that belong to both Beatty sequences [𝛼𝑛], [𝛽𝑚], and the numbers 1, 1 𝛼 , 1 𝛽 are linearly independent over the field of rational numbers, then 𝑆 𝛼,𝛽 (𝑁 ) ∼ 𝑁 𝛼𝛽 for 𝑁 → ∞.In this paper, we prove a refinement of this result for the case of algebraic numbers. Let 𝛼, 𝛽 > 1 be irrational algebraic numbers such that 1, 1 𝛼 , 1 𝛽 are linearly independent over the field of rational numbers. Then for any 𝜀 > 0 the following asymptotic formula holds:𝑆 𝛼,𝛽 (𝑁 ) = 𝑁 𝛼𝛽 + 𝑂 (︀ 𝑁 1 2 +𝜀 )︀ , 𝑁 → ∞.

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