Abstract
Gelfond proved that for coprime š ā 1 and š sums of digits of š-ary expressions of natural numbers are uniformly distributed on arithmetic progressions with the common difference š. Later, similar result was proved for the representations of natural numbers based on linear recurrent sequences. We consider the remainder term of the corresponding asymptotic formula and study the dichotomy between the logarithmic and power estimates of the remainder term. In the case š = 2, we obtain some sufficient condition for the validity of the logarithmic estimate. Using them, we show that the logarithmic estimate holds for expansions based on all second-order linear recurrenct sequences and on infinite family of third-order sequences. Also we construct an example of the linear reccurent sequence of an arbitrary order with such property. On the other hand, we give an example of a third-order linear recurrent sequence for which the logarithmic estimate does not hold. We also show that for š = 3 the logarithmic estimate does not hold even in the simplest case of the expansions based on Fibonacci numbers. In addition, we consider the representations of natural numbers based on the denominators of partial convergents of the continued fraction expansions of irrational numbers. In this case, we prove the uniformity of the distribution of sums of digits over arithmetic progressions with the common difference 2 with the logarithmic remainder term.
Highlights
We prove the uniformity of the distribution of sums of digits over arithmetic progressions with the common difference 2 with the logarithmic remainder term
ŠŃŠøŠ²ŠµŠ“ŠµŠ½Š½Š¾Šµ Š“Š¾ŠŗŠ°Š·Š°ŃŠµŠ»ŃŃŃŠ²Š¾ ŠæŠ¾ŠŗŠ°Š·ŃŠ²Š°ŠµŃ ŃŠ°ŠŗŠ¶Šµ, ŃŃŠ¾ ŠµŃŠ»Šø ŠæŠ¾ŃŠ»ŠµŠ“Š¾Š²Š°ŃŠµŠ»ŃŠ½Š¾ŃŃŃ Š·Š½Š°Š¼ŠµŠ½Š°ŃŠµŠ»ŠµŠ¹ ŠæŠ¾Š“Ń Š¾Š“ŃŃŠøŃ Š“ŃŠ¾Š±ŠµŠ¹ Šŗ Ī± ŃŠ°ŃŃŠµŃ Š±ŃŃŃŃŠµŠµ, ŃŠµŠ¼ ŃŠŗŃŠæŠ¾Š½ŠµŠ½ŃŠøŠ°Š»ŃŠ½Š¾, ŃŠ¾ Š¾ŃŠµŠ½ŠŗŃ ŠøŠ· ŃŠµŠ¾ŃŠµŠ¼Ń 6 Š¼Š¾Š¶Š½Š¾ ŃŠ»ŃŃŃŠøŃŃ
Š. Š ŃŃŠµŠ“Š½ŠøŃ Š·Š½Š°ŃŠµŠ½ŠøŃŃ ŃŃŠ½ŠŗŃŠøŠø Ļk(n) Š² Š½ŠµŠŗŠ¾ŃŠ¾ŃŃŃ ŠæŠ¾ŃŠ»ŠµŠ“Š¾Š²Š°ŃŠµŠ»ŃŠ½Š¾ŃŃŃŃ Š½Š°ŃŃŃŠ°Š»ŃŠ½ŃŃ ŃŠøŃŠµŠ» // ŠŠ°ŃŠµŠ¼
Summary
ŠŠµŠ»ŃŃŠ¾Š½Š“ Š“Š¾ŠŗŠ°Š·Š°Š» ŃŃŠ¾ ŠæŃŠø ŃŃŠ»Š¾Š²ŠøŠø Š²Š·Š°ŠøŠ¼Š½Š¾Š¹ ŠæŃŠ¾ŃŃŠ¾ŃŃ b ā 1 Šø d ŃŃŠ¼Š¼Ń ŃŠøŃŃ ŃŠ°Š·Š»Š¾Š¶ŠµŠ½ŠøŠ¹ Š½Š°ŃŃŃŠ°Š»ŃŠ½ŃŃ ŃŠøŃŠµŠ» Š² b-ŠøŃŠ½ŃŃ ŃŠøŃŃŠµŠ¼Ń ŃŃŠøŃŠ»ŠµŠ½ŠøŃ ŃŠ°Š²Š½Š¾Š¼ŠµŃŠ½Š¾ ŃŠ°ŃŠæŃŠµŠ“ŠµŠ»ŠµŠ½Ń ŠæŠ¾ Š°ŃŠøŃŠ¼ŠµŃŠøŃŠµŃŠŗŠøŠ¼ ŠæŃŠ¾Š³ŃŠµŃŃŠøŃŠ¼ Ń ŃŠ°Š·Š½Š¾ŃŃŃŃ d. ŠŠ¾Š·Š“Š½ŠµŠµ Š°Š½Š°Š»Š¾Š³ŠøŃŠ½ŃŠ¹ ŃŠµŠ·ŃŠ»ŃŃŠ°Ń Š±ŃŠ» ŠæŠ¾Š»ŃŃŠµŠ½ Š“Š»Ń ŃŠ°Š·Š»Š¾Š¶ŠµŠ½ŠøŠ¹ Š½Š°ŃŃŃŠ°Š»ŃŠ½ŃŃ ŃŠøŃŠµŠ» ŠæŠ¾ Š»ŠøŠ½ŠµŠ¹Š½ŃŠ¼ ŃŠµŠŗŃŃŃŠµŠ½ŃŠ½ŃŠ¼ ŠæŠ¾ŃŠ»ŠµŠ“Š¾Š²Š°ŃŠµŠ»ŃŠ½Š¾ŃŃŃŠ¼. Š” ŠøŃ ŠæŠ¾Š¼Š¾ŃŃŃ ŠæŠ¾ŠŗŠ°Š·Š°Š½Š¾, ŃŃŠ¾ Š»Š¾Š³Š°ŃŠøŃŠ¼ŠøŃŠµŃŠŗŠ°Ń Š¾ŃŠµŠ½ŠŗŠ° ŠøŠ¼ŠµŠµŃ Š¼ŠµŃŃŠ¾ Š“Š»Ń ŃŠ°Š·Š»Š¾Š¶ŠµŠ½ŠøŠ¹ ŠæŠ¾ Š²ŃŠµŠ¼ ŃŠµŠŗŃŃŃŠµŠ½ŃŠ½ŃŠ¼ ŠæŠ¾ŃŠ»ŠµŠ“Š¾Š²Š°ŃŠµŠ»ŃŠ½Š¾ŃŃŃŠ¼ ŠæŠ¾ŃŃŠ“ŠŗŠ° 2 Šø Š±ŠµŃŠŗŠ¾Š½ŠµŃŠ½Š¾Š¼Ń ŃŠµŠ¼ŠµŠ¹ŃŃŠ²Ń ŠæŠ¾ŃŠ»ŠµŠ“Š¾Š²Š°ŃŠµŠ»ŃŠ½Š¾ŃŃŠµŠ¹ ŠæŠ¾ŃŃŠ“ŠŗŠ° 3, Š° ŃŠ°ŠŗŠ¶Šµ ŃŃŃŠ¾ŠøŠ¼ ŠæŃŠøŠ¼ŠµŃ Š»ŠøŠ½ŠµŠ¹Š½Š¾Š¹ ŃŠµŠŗŃŃŃŠµŠ½ŃŠ½Š¾Š¹ ŠæŠ¾ŃŠ»ŠµŠ“Š¾Š²Š°ŃŠµŠ»ŃŠ½Š¾ŃŃŠø ŠæŃŠ¾ŠøŠ·Š²Š¾Š»ŃŠ½Š¾Š³Š¾ ŠæŠ¾ŃŃŠ“ŠŗŠ° Ń ŃŠ°ŠŗŠøŠ¼ ŃŠ²Š¾Š¹ŃŃŠ²Š¾Š¼. Š” Š“ŃŃŠ³Š¾Š¹ ŃŃŠ¾ŃŠ¾Š½Ń, Š¼Ń ŠæŃŠøŠ²Š¾Š“ŠøŠ¼ ŠæŃŠøŠ¼ŠµŃ Š»ŠøŠ½ŠµŠ¹Š½Š¾Š¹ ŃŠµŠŗŃŃŃŠµŠ½ŃŠ½Š¾Š¹ ŠæŠ¾ŃŠ»ŠµŠ“Š¾Š²Š°ŃŠµŠ»ŃŠ½Š¾ŃŃŠø ŃŃŠµŃŃŠµŠ³Š¾ ŠæŠ¾ŃŃŠ“ŠŗŠ°, Š“Š»Ń ŠŗŠ¾ŃŠ¾ŃŠ¾Š¹ Š»Š¾Š³Š°ŃŠøŃŠ¼ŠøŃŠµŃŠŗŠ°Ń Š¾ŃŠµŠ½ŠŗŠ° Š½Šµ ŠøŠ¼ŠµŠµŃ Š¼ŠµŃŃŠ°. Š§ŃŠ¾ Š“Š»Ń d = 3 Š»Š¾Š³Š°ŃŠøŃŠ¼ŠøŃŠµŃŠŗŠ°Ń Š¾ŃŠµŠ½ŠŗŠ° Š½Šµ ŠøŠ¼ŠµŠµŃ Š¼ŠµŃŃŠ° ŃŠ¶Šµ Š² ŠæŃŠ¾ŃŃŠµŠ¹ŃŠµŠ¼ ŃŠ»ŃŃŠ°Šµ ŃŠ°Š·Š»Š¾Š¶ŠµŠ½ŠøŠ¹ ŠæŠ¾ ŃŠøŃŠ»Š°Š¼ Š¤ŠøŠ±Š¾Š½Š°ŃŃŠø. ŠŠ»ŃŃŠµŠ²ŃŠµ ŃŠ»Š¾Š²Š°: ŃŠøŃŠ»Š° Š¤ŠøŠ±Š¾Š½Š°ŃŃŠø, Š¾Š±Š¾Š±ŃŠµŠ½Š½ŃŠµ ŃŠ°Š·Š»Š¾Š¶ŠµŠ½ŠøŃ Š¦ŠµŠŗŠŗŠµŠ½Š“Š¾ŃŃŠ°, Š»ŠøŠ½ŠµŠ¹Š½ŃŠµ ŃŠµŠŗŃŃŃŠµŠ½ŃŠ½ŃŠµ ŠæŠ¾ŃŠ»ŠµŠ“Š¾Š²Š°ŃŠµŠ»ŃŠ½Š¾ŃŃŠø, ŃŠµŠæŠ½ŃŠµ Š“ŃŠ¾Š±Šø, ŃŃŠ¼Š¼Ń ŃŠøŃŃ, Š·Š°Š“Š°ŃŠ° ŠŠµŠ»ŃŃŠ¾Š½Š“Š°.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have