@article{Li2025, 
author = {Xinyan Li and Wenxu Ge},
title = {A Diophantine approximation problem with unlike powers of primes},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {1},
pages = {736-753},
keywords = {primes, Diophantine inequality, exceptional set, Davenport-Heilbronn method},
url = {https://www.sciopen.com/article/10.3934/math.2025034},
doi = {10.3934/math.2025034},
abstract = {Let        λ          1      ,        λ          2      ,        λ          3      , and        λ          4       be non-zero real numbers, not all negative. Suppose that                                λ                          1                          /                                λ                          3                      is irrational and algebraic,    δ  >  0, and the set        V   is a well-spaced sequence. In this paper, we prove that, for any    ε  >  0, the number of    v  ∈      V   with    v  ≤  X such that the inequality                                 |                                                                      λ                                                  1                                                                                                      p                                                  1                                                      +                                                            λ                                                  2                                                                                                                          p                                                        2                                                                              2                                      +                                                            λ                                                  3                                                                                                                          p                                                        3                                                                              3                                      +                                                            λ                                                  4                                                                                                                          p                                                        4                                                                              4                                      −            v                    |                <                                            v                                      −              δ                                          has no solution in primes                      p                    1            ,                      p                    2            ,                      p                    3            ,                      p                    4             that does not exceed    O  (                    X                    1        −                  83          144                +        2        δ        +        2        ε              ).}
}