On some analogue of the Gelfond problem for Zeckendorf representations

Abstract

A.O. Gelfond proved that if 𝑏−1 and 𝑑 are coprime, the sums of digits of the 𝑏-ary expressions of natural numbers are uniformly distributed over arithmetic progressions with difference 𝑑. He also obtained a power estimate for the remainder term in this problem.
We consider an analogue of Gelfond’s problem for Zeckendorf representations of naturals as a sum of Fibonacci numbers. It is shown that in this case we again have the uniform distribution of the sums of digits over arithmetic progressions.
Moreover, in the case when the difference of the arithmetic progression 𝑑 is equal to 2, it was previously proved that the remainder term of the problem is logarithmic. In the present paper, it is shown that for 𝑑 ≥ 3 the remainder term of the problem is a power and an unimprovable in order estimate for it is found.
The proof is based on the detailed study of the remainder term at the Fibonacci numbers. It is shown that the remainder term at an arbitrary point can be estimated through the values of the remainder term in points equal to Fibonacci numbers. For them, it is possible to obtain a linear recurrence relation with constant coefficients, and, moreover, and an exact formula in terms of some Vandermonde determinants connected with the roots of the characteristic
polynomial.
Moreover, quite surprisingly, the linear recurrence relation for the remainder term at the Fibonacci points turns out to be connected with some combinatorial triangles, similar to Pascal’s triangle.