Binary and ternary congruences involving intervals and sets modulo a prime

2024/10/05 by Garaev, Moubariz Z., Pardo, Julio C., Shparlinski, Igor E.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2410.03991

Abstract

Let s be a fixed positive integer constant, ε be a fixed small positive number. Then, provided that a prime p is large enough, we prove that for any set \\mathcal M⊆ \mathbb F_p^* of size |\mathcal M|= \lfloor p^14/29\rfloor and integer H=\lfloor p^14/29+ε\rfloor, any integer λ can be represented in the form (m₁)/(x₁^s)+(m₂)/(x₂^s)+(m₃)/(x₃^s)≡ λ\bmod p, with m_i∈ \mathcal M, 1≤ x_i≤ H, i=1,2,3. When s=1 we show that for almost all primes p the following holds: if |\mathcal M|= \lfloor p^1/2\rfloor and H=\lfloor p^1/2(log p)^6+ε\rfloor, then any integer λ can be represented in the form (m₁)/(x₁)+(m₂)/(x₂)≡ λ\bmod p, with m_i∈ \mathcal M, 1≤ x_i≤ H, i=1,2.

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• Multiplicative Congruences with Variables from Short Intervals
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• Short intervals containing a prescribed number of primes
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