The Distribution of Fractional Parts with Applications to Gap Results in Number Theory

On the mean value of the index of composition of an integer

“…This is Lemma 3.3 of Zhai [5]. The exponent 1/(2d + 1), which is connected with gap results for d-free numbers, is due to Filaseta and Trifonov [3]. Now we prove Proposition A.…”

On primitive lattice points in planar domains

“…For this sum, we have the following Lemma 7.1, which is contained in the proof of Theorem 1 of Filaseta and Trifonov [2].…”

A note on a multiplicative hybrid problem

“…In the one-dimensional case, upper bounds of such numbers can be obtained by using results dealing with divided differences (see [2,4]). …”