On erdos and turan conjecture on arithmetic progressions

A concise proof of Erdos and Turan conjecture on arithmetic progressions is given in this paper. Whether the set A⊂{1,2,⋯,N} (N≥2) contains no non-trivial arithmetic progressions is equivalent to whether the corresponding linear Diophantine equation has no integer solutions, so the combinatorial approximation method to obtain the approximate number of integer solutions to the linear Diophantine equation with integer subsets as solution set is introduced. In different situations (Bloom and Sisask's situation, Behrend's situation, the case of powers of integer), we assume different forms or models of density of the progressions or the integer subsets to compute the approximate number of integer solutions to the corresponding linear Diophantine equation. When the corresponding linear Diophantine equation has no integer solutions, we can get the density of progressions. The approximate number of integer solutions of corresponding linear Diophantine equations of the 2-term and 3-term arithmetic progressions and the density of the 2-term and 3-term arithmetic progressions are computed to validate the combinatorial approximation method. The results of 2-term and 3-term arithmetic progressions are improved and the results of n-term arithmetic progressions are obtained.