Bound of character sums associated with Beatty sequences

This study is on the bound problems of Beatty sequences. Beatty sequences appear with special versatility in the arithmetic properties of sequences which is in the form of [αᵣ] where α is an irrational and r is a natural number. This study consider nonhomogeneous Beatty sequences in the set of Bα,Bα = {[αᵣ + β] : r = 1,2,3,...} and concerns on solving distribution of bounds sequences with different conditions of integer parts r. The estimation of the double character sums can be obtained by identifying the cardinality of double character sums. The identification of cardinality applies properties of character sums associated with composite moduli. It consists of multiplicative and additive character sums with representation of polar form in double character sums extended to composite moduli. The character sums in this form ∑X X(d1)X(d2) gives the result rely on φ(m)+1 if d1 is equal to d2 and for the rest conditions will give zeroes. Thus the cardinality of double character sums obtained is much less than φ(m)R#K where R is the highest integer terms in the sequences under consideration. Discrepancy of the sequences is used to estimate the bound due to the ability of measuring the uniformity of the sequences. Then, by applying discrepancy, Cauchy inequalities and the cardinality of the double character sums associated with composite moduli, the estimation of Beatty sequences with different conditions of integral parts r is obtained. This study provide six different conditions of integral part r which are r is prime number associated with prime modulo, r is prime number associated with composite moduli, r is Fibonacci number associated with composite moduli, r is y-smooth number associated with prime modulo, r is y-smooth number associated with composite moduli and r is natural number associated with composite moduli. In general, the result of bound problem under consideration is much less than (φ(m)) ¼ R+RDαβ (R). As a conclusion the bounds of the character sums of Beatty sequences are depends on the size of cardinality of double character sums.

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