Sign changes of Kloosterman sums with almost prime moduli

We prove that the Kloosterman sum $$S(1,1;c)$$ S ( 1 , 1 ; c ) changes sign infinitely often as $$c$$ c runs over squarefree moduli with at most 10 prime factors, which improves the previous results of Fouvry and Michel, Sivak-Fischler and Matomäki, replacing 10 by 23, 18 and 15, respectively. The method combines the Selberg sieve, equidistribution of Kloosterman sums and spectral theory of automorphic forms