Engel Series and Cohen-Egyptian Fraction Expansions

Abstract

Two kinds of series representations, referred to as the Engel series and the Cohen-Egyptian fraction expansions, of elements in two different fields, namely, the real number and the discrete-valued non-archimedean fields are constructed. Both representations are shown to be identical in all cases except the case of real rational numbers.

1. Introduction

It is well known [1] that each nonzero real number can be uniquely written as an Engel series expansion, or ES expansion for short, and an ES expansion represents a rational number if and only if each digit in such expansion is identical from certain point onward. In 1973, Cohen [2] devised an algorithm to uniquely represent each nonzero real number as a sum of Egyptian fractions, which we refer to as its Cohen-Egyptian fraction (or CEF) expansion. Cohen also characterized the real rational numbers as those with finite CEF expansions. At a glance, the shapes of both expansions seem quite similar. This naturally leads to the question whether the two expansions are related. We answer this question affirmatively for elements in two different fields. In Section 2, we treat the case of real numbers and show that for irrational numbers both kinds of expansion are identical, while for rational numbers, their ES expansions are infinite, periodic of period 1, but their CEF expansions always terminate. In Section 3, we treat the case of a discrete-valued non-archimedean field. After devising ES and CEF expansions for nonzero elements in this field, we see immediately that both expansions are identical. In Section 4, we characterize rational elements in three different non-archimedean fields.

2. The Case of Real Numbers

Recall the following result, see, for example, Kapitel IV of [1], which asserts that each nonzero real number can be uniquely represented as an infinite ES expansion and rational numbers have periodic ES expansions of period 1.

Theorem 2.1. Each 𝐴∈ℝ⧵{0} is uniquely representable as an infinite series expansion, called its Engel series (ES) expansion, of the form 𝐴=𝑎₀+^∞_𝑛=11𝑎₁𝑎₂⋯𝑎_𝑛, (2.1) where 𝑎₀=𝑎[𝐴],if𝐴∉ℤ,𝐴−1if𝐴∈ℤ,¹≥2,𝑎_𝑛+1≥𝑎_𝑛(𝑛≥1). (2.2) Moreover, 𝐴∈ℚ if and only if 𝑎_𝑛+1=𝑎_𝑛(≥2) for all sufficiently large 𝑛.

Proof. Define 𝐴₁=𝐴−𝑎₀, then 0<𝐴₁≤1. If 𝐴_𝑛≠0(𝑛≥1) is already defined, put 𝑎_𝑛1=1+[𝐴_𝑛]𝐴_𝑛+1,=𝑎_𝑛𝐴_𝑛−1. (2.3)(2.4) Observe that 𝑎_𝑛 is the least integer >1/𝐴_𝑛 and 1𝑎_𝑛<𝐴_𝑛≤1𝑎_𝑛.−1 (2.5) We now prove the folowing.Claim 2. We have 0<⋯≤𝐴_𝑛+1≤𝐴_𝑛≤⋯≤𝐴₂≤𝐴₁≤1.Proof. First, we show that 𝐴_𝑛>0 for all 𝑛≥1 by induction. If 𝑛=1, we have seen that 𝐴₁>0. Assume now that 𝐴_𝑛>0 for 𝑛≥1. By (2.3), we see that 𝑎_𝑛∈ℕ. Since 𝐴_𝑛+1=𝑎_𝑛𝐴_𝑛−1=(𝐴_𝑛−1𝑎_𝑛)𝑎_𝑛 (2.6) and 1/𝑎_𝑛<𝐴_𝑛, we have 𝐴_𝑛+1>0. If there exists 𝑘∈ℕ such that 𝐴_𝑘+1>𝐴_𝑘, then 𝑎_𝑘𝐴_𝑘−1=𝐴_𝑘+1>𝐴_𝑘 (2.7) and so 𝑎_𝑘−1>1/𝐴_𝑘, contradicting the minimal property of 𝑎_𝑛 and the Claim is proved.From the Claim and (2.3), we deduce that 𝑎₁≥2 and 𝑎_𝑛+1≥𝑎_𝑛(𝑛≥1). Iterating (2.4), we get 𝐴₁=1𝑎₁+1𝑎₁𝑎₂1+⋯+𝑎₁𝑎₂⋯𝑎_𝑛+𝐴_𝑛+1𝑎₁𝑎₂⋯𝑎_𝑛. (2.8) To establish convergence, let 𝐵_𝑛=1𝑎₁+1𝑎₁𝑎₂1+⋯+𝑎₁𝑎₂⋯𝑎_𝑛(𝑛≥1). (2.9) Since 𝐴_𝑛>0 and 𝑎_𝑛∈ℕ for all 𝑛≥1, the sequence of real numbers (𝐵_𝑛) is increasing and bounded above by 𝐴₁. Thus, lim_𝑛→∞𝐵_𝑛 exists and so 1𝑎₁𝑎₂⋯𝑎_𝑛→0(𝑛→∞). (2.10) By the Claim, 𝐴0<^𝑛+1𝑎₁𝑎₂⋯𝑎_𝑛≤1𝑎₁𝑎₂⋯𝑎_𝑛→0(𝑛→∞), (2.11) showing that any real number has an ES expansion. To prove uniqueness, assume that we have two infinite such expansions such that 𝑎₀+^∞_𝑛=11𝑎₁𝑎₂⋯𝑎_𝑛=𝑏₀+^∞_𝑛=11𝑏₁𝑏₂⋯𝑏_𝑛, (2.12) with the restrictions 𝑎₀∈ℤ,𝑎₁≥2,𝑎_𝑛+1≥𝑎_𝑛(𝑛≥1) and the same restrictions also for the 𝑏_𝑛's. From the restrictions, we note that 0<𝐴₁∶=^∞_𝑛=11𝑎₁𝑎₂⋯𝑎_𝑛≤1. (2.13) If 𝐴₁=1, then by (2.12) we also have ∑_𝑛≥11/𝑏₁𝑏₂⋯𝑏_𝑛=1, forcing 𝑎₀=𝑏₀. If 0<𝐴₁<1, then (2.12) shows that 0<∑_𝑛≥11/𝑏₁𝑏₂⋯𝑏_𝑛<1, forcing again 𝑎₀=𝑏₀. In either case, cancelling out the terms 𝑎₀,𝑏₀ in (2.12) we get 𝐴₁∶=^∞_𝑛=11𝑎₁𝑎₂⋯𝑎_𝑛=^∞_𝑛=11𝑏₁𝑏₂⋯𝑏_𝑛. (2.14) Since 𝑎_𝑛+1≥𝑎_𝑛, then 𝑎₁𝐴₁1−1=𝑎₂+1𝑎₂𝑎₃+1𝑎₂𝑎₃𝑎₄1+⋯≤𝑎₁+1𝑎₁𝑎₂+1𝑎₁𝑎₂𝑎₃+⋯=𝐴₁, (2.15) so 0<𝑎₁−1/𝐴₁≤1. But there is exactly one integer 𝑎₁ satisfying these restrictions. Thus, 𝑎₁=𝑏₁. Cancelling out the terms 𝑎₁ and 𝑏₁ in (2.14) and repeating the arguments we see that 𝑎_𝑖=𝑏_𝑖 for all 𝑖.
Concerning the rationality characterization, if its ES expansion is infinite periodic of period 1, it clearly represents a rational number. To prove its converse, let 𝐴=𝑎/𝑏∈ℚ⧵{0}. Since
𝐴₁=𝐴−𝑎₀=𝑎−𝑏𝑎₀𝑏, (2.16) we see that 𝐴₁ is a rational number in the interval (0,1] whose denominator is 𝑏. In general, from (2.4), we deduce that each 𝐴_𝑛(𝑛≥1) is a rational number in the interval (0,1] whose denominator is 𝑏. But the number of rational numbers in the interval (0,1] whose denominator is 𝑏 is finite implying that there are two least suffixes ℎ,𝑘∈ℕ such that 𝐴_ℎ+𝑘=𝐴_ℎ. Thus, by (2.3), we have 𝑎_ℎ+𝑘=𝑎_ℎ. From (2.2), we know that the sequence {𝑎_𝑛} is increasing. We must then have 𝑘=1 and the assertion follows.

Remark 2 s. In passing, we make the following observations. (a)For 𝑛≥1, we have 𝑎_𝑛+1=𝑎_𝑛⟸𝐴_𝑛+1=𝐴_𝑛⇔𝑎_𝑛𝐴_𝑛−1=𝐴_𝑛⇔𝑎_𝑛1=1+𝐴_𝑛⇔1𝐴_𝑛∈ℤ. (2.17)(b)If 𝐴∈ℚ^𝑐, then 𝐴_𝑛∈ℚ^𝑐 and so 1/𝐴_𝑛∉ℚ for all 𝑛≥1. (c)If 𝐴∈ℤ, then its ES expansion is 1𝐴=𝐴−1+2+12²+12³+⋯. (2.18)

To construct a Cohen-Egyptian fraction expansion, we proceed as in [2] making use of the following lemma.

Lemma 2.2. For any 𝑦∈(0,1), there exist a unique integer 𝑛≥2 and a unique 𝑟∈ℝ such that 1=𝑛𝑦−𝑟,0≤𝑟<𝑦. (2.19)

Proof. Let 𝑛=⌈1/𝑦⌉∈ℕ and 𝑟=𝑛𝑦−1. Put ⟨1/𝑦⟩∶=𝑛−1/𝑦∈[0,1) and so 1𝑟=𝑛𝑦−1=𝑦⟨𝑦⟩∈[0,𝑦). (2.20) To prove uniqueness, assume 𝑛𝑦−𝑟=1=𝑚𝑦−𝑠 so that 11+𝑦>𝑛=1+𝑟𝑦≥1𝑦. (2.21) Since there is only one integer with this property, we deduce 𝑛=𝑚 and consequently, 𝑟=𝑠 proving the lemma.

Theorem 2.3. Each 𝐴∈ℝ⧵{0} is uniquely representable as a CEF expansion of the form 𝐴=𝑛₀+^∞_𝑘=11𝑛₁𝑛₂⋯𝑛_𝑘, (2.22) subject to the condition 𝑛₀∈ℤ,𝑛₁≥2,𝑛_𝑘+1≥𝑛_𝑘(𝑘≥1), (2.23) and no term of the sequence appears infinitely often. Moreover, each CEF expansion terminates if and only if it represents a rational number.

Proof. To construct a CEF expansion for 𝐴∈ℝ⧵{0}, define 𝑟₀=𝐴−𝑛₀∈[0,1). (2.24) If 𝑟₀=0, then the process stops and we write 𝐴=𝑛₀. If 𝑟₀≠0, by Lemma 2.2, there are unique 𝑛₁∈ℕ and 𝑟₁∈ℝ such that 1=𝑛₁𝑟₀−𝑟₁,0≤𝑟₁<𝑟₀,𝑛₁≥2. (2.25) Thus, 𝐴=𝑛₀+𝑟₀=𝑛₀+1𝑛₁+𝑟₁𝑛₁. (2.26) If 𝑟₁=0, then the process stops and we write 𝐴=𝑛₀+1/𝑛₁. If 𝑟₁≠0, by Lemma 2.2, there are 𝑛₂∈ℕ and 𝑟₂∈ℝ such that 1=𝑛₂𝑟₁−𝑟₂,0≤𝑟₂<𝑟₁,𝑛₂≥𝑛₁, (2.27) the last inequality being followed from 𝑛₁=⌈1/𝑟₀⌉,𝑛₂=⌈1/𝑟₁⌉, and 𝑟₁<𝑟₀. Observe also that 𝐴=𝑛₀+1𝑛₁+1𝑛₁𝑛₂+𝑟₂𝑛₁𝑛₂. (2.28) Continuing this process, we get 𝐴=𝑛₀+1𝑛₁+1𝑛₁𝑛₂1+⋯+𝑛₁𝑛₂⋯𝑛_𝑘+𝑟_𝑘𝑛₁𝑛₂⋯𝑛_𝑘, (2.29) with 1=𝑛_𝑖𝑟_𝑖−1−𝑟_𝑖,1>𝑟_𝑖−1>𝑟_𝑖≥0,2≤𝑛_𝑖≤𝑛_𝑖+1(𝑖=1,2,…). (2.30) If some 𝑟_𝑘=0, then the process stops, otherwise the series convergence follows at once from |𝑟_𝑘𝑛₁𝑛₂⋯𝑛_𝑘|→0(𝑘→∞). (2.31)
To prove uniqueness, let
𝑛₀+^∞_𝑘=11𝑛₁𝑛₂⋯𝑛_𝑘=𝐴=𝑚₀+^∞_𝑘=11𝑚₁𝑚₂⋯𝑚_𝑘, (2.32) with the restrictions (2.23) on both digits 𝑛_𝑖 and 𝑚_𝑗. Now _𝑘≥11𝑛₁𝑛₂⋯𝑛_𝑘≤_𝑘≥112^𝑘=1. (2.33) It is clear that the restrictions (2.23) imply the strict inequality in (2.33). This also applies to the right-hand sum in (2.32). Equating integer and fractional parts in (2.32), we get 𝑛₀=𝑚₀,^∞_𝑘=11𝑛₁𝑛₂⋯𝑛_𝑘=^∞_𝑘=11𝑚₁𝑚₂⋯𝑚_𝑘=∶𝑤,say. (2.34) Since 𝑛_𝑘+1≥𝑛_𝑘, then 𝑛₁1𝑤−1=𝑛₂+1𝑛₂𝑛₃+1𝑛₂𝑛₃𝑛₄1+⋯≤𝑛₁+1𝑛₁𝑛₂+1𝑛₁𝑛₂𝑛₃+⋯=𝑤, (2.35) so 0<𝑛₁−1/𝑤≤1. But there is exactly one integer 𝑛₁ satisfying these restrictions. Then 𝑛₁=𝑚₁ and _𝑘≥21𝑛₂⋯𝑛_𝑘=_𝑘≥21𝑚₂⋯𝑚_𝑘. (2.36) Proceeding in the same manner, we conclude that 𝑛_𝑖=𝑚_𝑖 for all 𝑖.
Finally, we look at its rationality characterization. If 𝐴∈ℚ, then 𝑟₀∈ℚ, say 𝑟₀∶=𝑝/𝑞, where 𝑝,𝑞∈ℕ. From (2.30), we see that each 𝑟_𝑖 is a rational number whose denominator is 𝑞. Using this fact and the second inequality condition in (2.30), we deduce that 𝑟_𝑗=0 for some 𝑗≤𝑝, that is, the expansion terminates. On the other hand, it is clear that each terminating CEF expansion represents a rational number. Now suppose that 𝐴 is irrational and there is a 𝑗 and integer 𝑛 such that 𝑛_𝑖=𝑛 for all 𝑖≥𝑗. Then
𝐴=𝑛₀+^𝑗_𝑘=11𝑛₁𝑛₂⋯𝑛_𝑘+1𝑛₁𝑛₂⋯𝑛_𝑗^∞_𝑘=11𝑛^𝑘. (2.37) Since ∑_𝑘≥11/𝑛^𝑘=1/(𝑛−1), it follows that 𝐴 is rational, which is impossible.

The connection and distinction between ES and CEF expansions of a real number are described in the next theorem.

Theorem 2.4. Let 𝐴∈ℝ⧵{0} and the notation be as set out in Theorems 2.1 and 2.3.
(i) If 𝐴∈ℚ, then its ES expansion is infinite periodic of period 1, while its CEF expansion is finite. More precisely, for 𝐴∈ℚ⧵ℤ, let its ES and CEF expansions be, respectively,
𝐴=𝑎₀+^∞_𝑛=11𝑎₁𝑎₂⋯𝑎_𝑛=𝑛₀+^∞_𝑘=11𝑛₁𝑛₂⋯𝑛_𝑘. (2.38) If 𝑚 is the least positive integer such that 1/𝐴_𝑚∈ℤ, then 𝑎₀=𝑛₀,𝑎₁=𝑛₁,…,𝑎_𝑚−1=𝑛_𝑚−1,𝑎_𝑚=𝑛_𝑚+1,𝑎_𝑚=𝑎_𝑚+𝑖(𝑖≥1), (2.39) and the digits 𝑛_𝑖 terminate at 𝑛_𝑚.
(ii) If 𝐴∈ℝ⧵ℚ, then its ES and its CEF expansions are identical.

Proof. Both assertions follow mostly from Theorems 2.1, 2.3, and Remark (b) except for the result related to the expansions in (2.38) which we show now.
Let 𝐴∈ℚ⧵ℤ and let 𝑚 be the least positive integer such that 1/𝐴_𝑚∈ℤ. We treat two seperate cases.
Case 1 (𝑚=1). In this case, we have 1/𝐴₁∈ℤ and 𝑎₁=1+[1/𝐴₁]=1+1/𝐴₁. Since 𝑟₀=𝐴−𝑛₀=𝐴−[𝐴]=𝐴−𝑎₀=𝐴₁, we get 𝑛₁=1/𝐴₁ and so 𝑎₁=𝑛₁+1. We have 𝑟₁=𝑛₁𝑟₀−1=0, and so the CEF expansion terminates. On the other hand, by Remark (a) after Theorem 2.1, we have 𝑎₁=𝑎_𝑖(𝑖≥2).Case 2 (𝑚>1). Thus, 1/𝐴₁∉ℤ and 𝐴₁=𝑟₀. By Lemma 2.2, we have 𝑎₁=𝑛₁. For 1≤𝑖≤𝑚−2, assume that 𝐴_𝑖=𝑟_𝑖−1 and 𝑎_𝑖=𝑛_𝑖. Then 𝐴_𝑖+1=𝑎_𝑖𝐴_𝑖−1=𝑛_𝑖𝑟_𝑖−1−1=𝑟_𝑖. (2.40) Since 1/𝐴_𝑖+1∉ℤ, again by Lemma 2.2, 𝑎_𝑖+1=𝑛_𝑖+1. This shows that 𝑎₁=𝑛₁,…,𝑎_𝑚−1=𝑛_𝑚−1. Since 1/𝐴_𝑚∈ℤ, we have 𝑎_𝑚=1+[1/𝐴_𝑚]=1+1/𝐴_𝑚 and thus 𝐴_𝑚=𝑎_𝑚−1𝐴_𝑚−1−1=𝑛_𝑚−1𝑟_𝑚−2−1=𝑟_𝑚−1. (2.41) From the construction of CEF, we know that 𝑛_𝑚=⌈1/𝑟_𝑚−1⌉. Thus, 𝑛_𝑚=1/𝐴_𝑚 showing that 𝑎_𝑚=𝑛_𝑚+1. Furthermore, 𝑟_𝑚=𝑛_𝑚𝑟_𝑚−1−1=0, implying that the CEF terminates at 𝑛_𝑚, and by Remark (a) after Theorem 2.1, 𝑎_𝑚=𝑎_𝑚+𝑖(𝑖≥1).

3. The Non-Archimedean Case

We recapitulate some facts about discrete-valued non-archimedean fields taken from [3, Chapter 4]. Let 𝐾 be a field complete with respect to a discrete non-archimedean valuation |⋅| and 𝒪∶={𝐴∈𝐾;|𝐴|≤1} its ring of integers. The set 𝒫∶={𝐴∈𝐾;|𝐴|<1} is an ideal in 𝒪 which is both a maximal ideal and a principal ideal generated by a prime element 𝜋∈𝐾. The quotient ring 𝒪/𝒫 is a field, called the residue class field. Let 𝒜⊂𝒪 be a set of representatives of 𝒪/𝒫. Every 𝐴∈𝐾⧵{0} is uniquely of the shape

𝐴=^∞_𝑛=𝑁𝑏_𝑛𝜋^𝑛(𝑏_𝑛∈𝒜,𝑏_𝑁≠0) (3.1) for some 𝑁∈ℤ, and define the order 𝑣(𝐴) of 𝐴 by |𝐴|=2^{−𝑣(𝐴)}=2^−𝑁, with 𝑣(0)∶=+∞. The head part ⟨𝐴⟩ of 𝐴 is defined as the finite series

⟨𝐴⟩=⁰_{𝑛=𝑣(𝐴)}𝑏_𝑛𝜋^𝑛if𝑣(𝐴)≤0,and0otherwise. (3.2) Denote the set of all head parts by

𝑆∶={⟨𝐴⟩;𝐴∈𝐾}. (3.3) The Knopfmachers' series expansion algorithm for series expansions in 𝐾 [4] proceeds as follows. For 𝐴∈𝐾, let 𝑎₀∶=⟨𝐴⟩∈𝑆. (3.4) Define 𝐴₁∶=𝐴−𝑎₀. (3.5) If 𝐴_𝑛≠0(𝑛≥1) is already defined, put

𝑎_𝑛1=⟨𝐴_𝑛⟩,𝐴_𝑛+1=(𝐴_𝑛−1𝑎_𝑛)𝑠_𝑛𝑟_𝑛 (3.6) if 𝑎_𝑛≠0, where 𝑟_𝑛 and 𝑠_𝑛∈𝐾⧵{0} which may depend on 𝑎₁,…,𝑎_𝑛. Then for 𝑛≥1

𝐴=𝑎₀+𝐴₁=⋯=𝑎₀+1𝑎₁+𝑟₁𝑠₁1𝑎₂𝑟+⋯+¹⋯𝑟_𝑛−1𝑠₁⋯𝑠_𝑛−11𝑎_𝑛+𝑟₁⋯𝑟_𝑛𝑠₁⋯𝑠_𝑛𝐴_𝑛+1. (3.7) The process ends in a finite expansion if some 𝐴_𝑛+1=0. If some 𝑎_𝑛=0, then 𝐴_𝑛+1 is not defined. To take care of this difficulty, we impose the condition

𝑣(𝑠_𝑛)−𝑣(𝑟_𝑛)≥2𝑣(𝑎_𝑛)−1. (3.8) Thus

𝐴=𝑎₀+1𝑎₁+^∞_𝑛=1𝑟₁⋯𝑟_𝑛𝑠₁⋯𝑠_𝑛⋅1𝑎_𝑛+1. (3.9) When 𝑟_𝑛=1,𝑠_𝑛=𝑎_𝑛, the algorithm yields a well-defined (with respect to the valuation) and unique series expansion, termed non-archimedean Engel series expansion. Summing up, we have the following.

Theorem 3.1. Every 𝐴∈𝐾⧵{0} has a finite or an infinite convergent non-archimedean ES expansion of the form 𝐴=𝑎₀+^∞_𝑛=11𝑎₁𝑎₂⋯𝑎_𝑛, (3.10) where the digits 𝑎_𝑛 are subject to the restrictions 𝑎₀=⟨𝐴⟩∈𝑆,𝑎_𝑛∈𝑆,𝑣(𝑎_𝑛)≤−𝑛,𝑣(𝑎_𝑛+1)≤𝑣(𝑎_𝑛)−1(𝑛≥1). (3.11)

Now we turn to the construction of a non-archimedean Cohen-Egyptian fraction expansion, in the same spirit as that of the real numbers, that is, by way of Lemma 2.2. To this end, we start with the following lemma.

Lemma 3.2. For any 𝑦∈𝐾⧵{0} such that 𝑣(𝑦)≥1, there exist a unique 𝑛∈𝑆 such that 𝑣(𝑛)≤−1 and a unique 𝑟∈𝐾 such that 1=𝑛𝑦−𝑟,𝑣(𝑟)≥𝑣(𝑦)+1(i.e.,0≤|𝑟|<|𝑦|). (3.12)

Proof. Let 𝑛=⟨1/𝑦⟩. Then 𝑣(𝑛)=𝑣(1/𝑦)=−𝑣(𝑦)≤−1. Putting 𝑟=𝑛𝑦−1, we show now that 𝑣(𝑟)≥𝑣(𝑦)+1. Since 𝑛=⟨1/𝑦⟩, we have 1𝑦=𝑛+_𝑘≥1𝑐_𝑘𝜋^𝑘, (3.13) where 𝑐_𝑘∈𝒜, and so 𝑛𝑦−1=−𝑦_𝑘≥1𝑐_𝑘𝜋^𝑘. (3.14) Thus 𝑣(𝑟)=𝑣(𝑛𝑦−1)=𝑣(−𝑦)+𝑣(_𝑘≥1𝑐_𝑘𝜋^𝑘)≥𝑣(𝑦)+1>𝑣(𝑦). (3.15) To prove the uniqueness, assume that there exist 𝑛₁∈𝑆 such that 𝑣(𝑛₁)≤−1 and 𝑟₁∈𝐾 such that 1=𝑛₁𝑦−𝑟₁,0≤|𝑟₁|<|𝑦|. (3.16) From 𝑛𝑦−𝑟=1=𝑛₁𝑦−𝑟₁, we get (𝑛−𝑛₁)𝑦=𝑟−𝑟₁. If 𝑛≠𝑛₁, since 𝑛,𝑛₁∈𝑆 we have |𝑛−𝑛₁|≥1. Using |𝑦|>|𝑟−𝑟₁|, we deduce that |𝑟−𝑟₁|<|𝑦|≤|𝑛−𝑛₁||𝑦|=|𝑟−𝑟₁|, (3.17) which is a contradiction. Thus, 𝑛=𝑛₁ and so 𝑟=𝑟₁.

For a non-archimedean CEF expansion, we now prove the following.

Theorem 3.3. Each 𝑦∈𝐾⧵{0} has a non-archimedean CEF expansion of the form 𝑦=𝑛₀+^∞_𝑘=11𝑛₁𝑛₂⋯𝑛_𝑘, (3.18) where 𝑛_𝑘∈𝑆,𝑣(𝑛_𝑘)≤−𝑘,𝑣(𝑛_𝑘+1)≤𝑣(𝑛_𝑘)−1(𝑘≥1). (3.19) This series representation is unique subject to the digit condition (3.19).

Proof. Define 𝑛₀=⟨𝑦⟩ and 𝑟₀=𝑦−𝑛₀. Then 𝑣(𝑟₀)≥1. If 𝑟₀=0, the process stops and we write 𝑦=𝑛₀. If 𝑟₀≠0, by Lemma 3.2, there are 𝑛₁∈𝑆 and 𝑟₁∈𝐾 such that 𝑛₁1=⟨𝑟₀⟩,𝑟₁=𝑛₁𝑟₀−1, (3.20) where 𝑣(𝑛₁)≤−1 and 𝑣(𝑟₁)≥𝑣(𝑟₀)+1. So 𝑦=𝑛₀+𝑟₀=𝑛₀+1𝑛₁+𝑟₁𝑛₁. (3.21) If 𝑟₁=0, the process stops and we write 𝑦=𝑛₀+1/𝑛₁. If 𝑟₁≠0, by Lemma 3.2, there are 𝑛₂∈𝑆 and 𝑟₂∈𝐾 such that 𝑛₂1=⟨𝑟₁⟩,𝑟₂=𝑛₂𝑟₁−1, (3.22) where 𝑣(𝑛₂)≤−1 and 𝑣(𝑟₂)≥𝑣(𝑟₁)+1. So 𝑦=𝑛₀+1𝑛₁+1𝑛₁𝑛₂+𝑟₂𝑛₁𝑛₂. (3.23) Continuing the process, in general, 𝑛_𝑘1=⟨𝑟_𝑘−1⟩,𝑟_𝑘=𝑛_𝑘𝑟_𝑘−1−1,𝑦=𝑛₀+1𝑛₁+1𝑛₁𝑛₂1+⋯+𝑛₁𝑛₂⋯𝑛_𝑘+𝑟_𝑘𝑛₁𝑛₂⋯𝑛_𝑘, (3.24) where 𝑛_𝑘∈𝑆,𝑣(𝑛_𝑘)≤−1,𝑣(𝑟_𝑘)≥𝑣(𝑟_𝑘−1)+1(𝑘≥1). (3.25) Thus, 𝑣(𝑛_𝑘+1)=−𝑣(𝑟_𝑘)≤−𝑣(𝑟_𝑘−1)−1=𝑣(𝑛_𝑘)−1(𝑘≥1). (3.26) We observe that the process terminates if 𝑟_𝑘=0. Next, we show that 𝑣(𝑛_𝑘)≤−𝑘(𝑘≥1). By construction, we have 𝑣(𝑛₁)≤−1. Assume that 𝑣(𝑛_𝑘)≤−𝑘, then 𝑣(𝑛_𝑘+1)≤𝑣(𝑛_𝑘)−1≤−𝑘−1. (3.27) Regarding convergence, consider 𝑟𝑣(^𝑘𝑛₁𝑛₂⋯𝑛_𝑘)=−𝑣(𝑛₁)−𝑣(𝑛₂)−⋯−𝑣(𝑛_𝑘)+𝑣(𝑟_𝑘)=−𝑣(𝑛₁)−𝑣(𝑛₂)−⋯−𝑣(𝑛_𝑘)−𝑣(𝑛_𝑘+1)≥1+2+⋯+𝑘+(𝑘+1)→∞(𝑘→∞). (3.28) It remains to prove the uniqueness. Suppose that 𝑥∈𝐾⧵{0} has two such expansions 𝑥=𝑛₀+_𝑗1𝑛₁𝑛₂⋯𝑛_𝑗=𝑚₀+_𝑖1𝑚₁𝑚₂⋯𝑚_𝑖. (3.29) Since 𝑣(∑_𝑗1/𝑛₁𝑛₂⋯𝑛_𝑗)=𝑣(1/𝑛₁)≥1 and 𝑛₀∈𝑆, we have 𝑛₀=⟨𝑦⟩. Similarly, 𝑚₀=⟨𝑦⟩ yielding by uniqueness 𝑛₀=𝑚₀ and ∑_𝑗≥11/𝑛₁𝑛₂⋯𝑛_𝑗=∑_𝑖≥11/𝑚₁𝑚₂⋯𝑚_𝑖. Putting 𝜔∶=_𝑗≥11𝑛₁𝑛₂⋯𝑛_𝑗=_𝑖≥11𝑚₁𝑚₂⋯𝑚_𝑖, (3.30) we have 𝑛₁𝜔=1+∑_𝑗≥21/𝑛₂⋯𝑛_𝑗 and so 1=𝑛₁𝜔−_𝑗≥21𝑛₂⋯𝑛_𝑗,𝑣(_𝑗≥21𝑛₂⋯𝑛_𝑗1)=𝑣(𝑛₂)=−𝑣(𝑛₂)≥−𝑣(𝑛₁)+1=𝑣(𝜔)+1. (3.31) By Lemma 3.2, since 𝑛₁ is the unique element in 𝑆 with such property, we deduce 𝑛₁=𝑚₁. Continuing in the same manner, we conclude that the two expansions are identical.

It is clear that the construction of non-archimedean ES and CEF expansions is identical which implies at once that the two representations are exactly the same in the non-archimedean case.

4. Rationality Characterization in the Non-Archimedean Case

In the case of real numbers, we have seen that both ES and CEF expansions can be used to characterize rational numbers with quite different outcomes. In the non-archimedean situation, though ES and CEF expansions are identical, their use to characterize rational elements depend significantly on the underlying nature of each specific field. We end this paper by providing information on the rationality characterization in three different non-archimedean fields, namely, the field of 𝑝-adic numbers and the two function fields, one completed with respect to the degree valuation and the other with respect to a prime-adic valuation.

The following characterization of rational numbers by 𝑝-adic ES expansions is due to Grabner and Knopfmacher [5].

Theorem 4.1. Let 𝑥∈𝑝ℤ_𝑝⧵{0}. Then 𝑥 is rational, 𝑥=𝛼/𝛽, if and only if either the 𝑝-adic ES expansion of 𝑥 is finite, or there exist an 𝑚 and an 𝑠≥𝑚, such that 𝑎_𝑚+𝑗=𝑝^𝑠+𝑗+1−𝛾𝑝^𝑠+𝑗(𝑗=0,1,2,…), (4.1) where 𝛾∣𝛽.

Now for function fields, we need more terminology. Let 𝔽 denote a field and 𝜋(𝑥) an irreducible polynomial of degree 𝑑 over 𝔽. There are two types of valuation in the field of rational functions 𝔽(𝑥), namely, the 𝜋(𝑥)-adic valuation |⋅|_𝜋, and the degree valuation |⋅|_1/𝑥 defined as follows. From the unique representation in 𝔽(𝑥),

𝑓(𝑥)𝑔(𝑥)=𝜋(𝑥)^𝑚𝑟(𝑥)𝑠(𝑥),𝑓(𝑥),𝑔(𝑥),𝑟(𝑥),𝑠(𝑥)∈𝔽[𝑥]⧵{0};𝜋(𝑥)∤𝑟(𝑥)𝑠(𝑥),𝑚∈ℤ, (4.2) set |0|_𝜋=0,|𝑓(𝑥)|𝑔(𝑥)_𝜋=2^−𝑚𝑑;|0|_1/𝑥=0,|𝑓(𝑥)|𝑔(𝑥)_1/𝑥=2^{deg𝑓(𝑥)−deg𝑔(𝑥)}. (4.3) Let 𝔽((𝜋)) and 𝔽((1/𝑥)) be the completions of 𝔽(𝑥), with respect to the 𝜋(𝑥)-adic and the degree valuations, respectively. The extension of the valuations to 𝔽((𝜋)) and 𝔽((1/𝑥)) is also denoted by |⋅|_𝜋 and |⋅|_1/𝑥.

For a characterization of rational elements, we prove the following.

Theorem 4.2. The CEF of 𝑦∈𝔽((𝜋)) or in 𝔽((1/𝑥)) terminates if and only if 𝑦∈𝔽(𝑥).

Proof. Although the assertions in both fields 𝔽((𝜋)) and 𝔽((1/𝑥)) are the same, their respective proofs are different. In fact, when the field 𝔽 has finite characteristic, both results have already been shown in [6] and the proof given here is basically the same.
We use the notation of the last section with added subscripts 𝜋 or 1/𝑥 to distinguish their corresponding meanings.
If the CEF of 𝑦 in either field is finite, then 𝑦 is clearly rational. It remains to prove the converse and we begin with the field 𝔽((𝜋)). Assume that 𝑦∈𝔽(𝑥)⧵{0}. By construction, each 𝑟_𝑘∈𝔽(𝑥) and so can be uniquely represented in the form
𝑟_𝑘=𝜋(𝑥)^{𝑣(𝑟𝑘)}𝑝_𝑘(𝑥)𝑞_𝑘,(𝑥) (4.4) where 𝑝_𝑘(𝑥),𝑞_𝑘(𝑥)(≠0)∈𝔽[𝑥] with gcd(𝑝_𝑘(𝑥),𝑞_𝑘(𝑥))=1,𝜋(𝑥)∤𝑝_𝑘(𝑥)𝑞_𝑘(𝑥). Since 𝑛_𝑘=⟨1/𝑟_𝑘−1⟩∈𝑆_𝜋 and 𝑣(𝑛_𝑘)≤−𝑘, it is of the form 𝑛_𝑘=𝑠_{𝑣(𝑛𝑘)}(𝑥)𝜋(𝑥)^{𝑣(𝑛𝑘)}+𝑠_{𝑣(𝑛𝑘)+1}(𝑥)𝜋(𝑥)^{𝑣(𝑛𝑘)+1}+⋯+𝑠₋₁(𝑥)𝜋(𝑥)⁻¹+𝑠₀(𝑥)=∶𝑚_𝑘(𝑥)𝜋(𝑥)^{𝑣(𝑛𝑘)}, (4.5) where 𝑠_{𝑣(𝑛𝑘)}(𝑥),…,𝑠₀(𝑥) are polynomials over 𝔽, not all 0, of degree <𝑑 and 𝑚_𝑘(𝑥)∈𝔽[𝑥]. Thus, |𝑛_𝑘|_1/𝑥≤max{|𝑠_{𝑣(𝑛𝑘)}(𝑥)𝜋(𝑥)^{𝑣(𝑛𝑘)}|_1/𝑥,|𝑠_{𝑣(𝑛𝑘)+1}(𝑥)𝜋(𝑥)^{𝑣(𝑛𝑘)+1}|_1/𝑥,…,|𝑠₋₁(𝑥)𝜋(𝑥)⁻¹|_1/𝑥,|𝑠₀(𝑥)|_1/𝑥}≤2^𝑑−1, (4.6) yielding |𝑚_𝑘(𝑥)|_1/𝑥≤2^{𝑑−𝑑𝑣(𝑛𝑘)−1}. (4.7) By construction, we have 𝑟_𝑘=𝑛_𝑘𝑟_𝑘−1−1. (4.8) Substituting (4.4) and (4.5) into (4.8) and using 𝑣(𝑟_𝑘−1)=−𝑣(𝑛_𝑘) lead to 𝜋(𝑥)^{−𝑣(𝑛𝑘+1)}𝑝_𝑘(𝑥)𝑞_𝑘−1(𝑥)=𝑞_𝑘(𝑥)(𝑚_𝑘(𝑥)𝑝_𝑘−1(𝑥)−𝑞_𝑘−1(𝑥)). (4.9) Since gcd(𝜋(𝑥)^{−𝑣(𝑛𝑘+1)}𝑝_𝑘(𝑥),𝑞_𝑘(𝑥))=1, it follows that 𝑞_𝑘(𝑥)∣𝑞_𝑘−1(𝑥), and so successively, we have |𝑞_𝑘(𝑥)|_1/𝑥≤|𝑞_𝑘−1(𝑥)|_1/𝑥≤⋯≤|𝑞₁(𝑥)|_1/𝑥, (4.10) which together with (4.9) yield |𝑝_𝑘(𝑥)|_1/𝑥≤|𝜋(𝑥)|^{𝑣(𝑛𝑘+1)}_1/𝑥max{|𝑚_𝑘(𝑥)𝑝_𝑘−1(𝑥)|_1/𝑥,|𝑞₁(𝑥)|_1/𝑥}. (4.11) Using (3.19) and (4.7), we consequently have |𝑝_𝑘(𝑥)|_1/𝑥≤2^{𝑑(𝑣(𝑛𝑘)−1)}max{2^{𝑑−𝑑𝑣(𝑛𝑘)−1}|𝑝_𝑘−1(𝑥)|_1/𝑥,|𝑞₁(𝑥)|_1/𝑥}1≤max{2|𝑝_𝑘−1(𝑥)|_1/𝑥,|𝑞₁(𝑥)|_1/𝑥2^𝑑(𝑘+1)}. (4.12) This shows that |𝑝_𝑘(𝑥)|_1/𝑥≤(1/2)|𝑝_𝑘−1(𝑥)|_1/𝑥 for all large 𝑘 which implies that from some 𝑘 onwards, 𝑝_𝑘(𝑥)=0, and so 𝑟_𝑘=0, that is, the expansion terminates.
Finally for the field 𝔽((1/𝑥)), assume that 𝑦=𝑝(𝑥)/𝑞(𝑥)∈𝔽(𝑥)⧵{0}. Without loss of generality, assume deg𝑝(𝑥)≥deg𝑞(𝑥). By the Euclidean algorithm, we have
𝑦=𝑝(𝑥)𝑞(𝑥)=𝑁₀𝑅(𝑥)+⁰(𝑥)𝑞(𝑥)∶=𝑛₀+𝑟₀, (4.13) where 𝑛₀∶=𝑁₀(𝑥)=⟨𝑦⟩_1/𝑥,𝑅₀(𝑥)∈𝔽[𝑥],0≤deg𝑅₀<deg𝑞,𝑟₀=𝑅₀(𝑥).𝑞(𝑥) (4.14) From the Euclidean algorithm, 𝑞(𝑥)𝑅₀(𝑥)=𝑁₁𝑅(𝑥)+¹(𝑥)𝑅₀(𝑥),𝑁₁(𝑥),𝑅₁(𝑥)∈𝔽[𝑥],0≤deg𝑅₁<deg𝑅₀<deg𝑞, (4.15) which is, in the terminology of Lemma 3.2, 1=𝑟₀𝑁₁+𝑅₁𝑞=𝑟₀𝑛₁−𝑟₁. (4.16) Again, from the Euclidean algorithm, −𝑞(𝑥)𝑅₁(𝑥)=𝑁₂𝑅(𝑥)+²(𝑥)𝑅₁(𝑥),𝑁₂(𝑥),𝑅₂(𝑥)∈𝔽[𝑥],0≤deg𝑅₂<deg𝑅₁<deg𝑅₀<deg𝑞, (4.17) which is, in the terminology of Lemma 3.2, 1=𝑟₁𝑁₂−𝑅₂𝑞=𝑟₁𝑛₂−𝑟₂. (4.18) Proceeding in the same manner, in general we have 𝑟_𝑗=(−1)^𝑗𝑅_𝑗𝑞,0≤deg𝑅_𝑗<deg𝑅_𝑗−1<⋯<deg𝑅₁<deg𝑞. (4.19) There must then exist 𝑘∈ℕ such that deg𝑅_𝑘=0, that is, 𝑅_𝑘∈𝔽⧵{0}. Thus, the CEF of 𝑦 is 𝑦=𝑛₀+1𝑛₁1+⋯+𝑛₁⋯𝑛_𝑘+𝑟_𝑘𝑛₁⋯𝑛_𝑘=𝑛₀+1𝑛₁1+⋯+𝑛₁⋯𝑛_𝑘+1𝑛₁⋯𝑛_𝑘𝑛_𝑘+1, (4.20) where 𝑛_𝑘+1=(−1)^𝑘𝑅_𝑘⁻¹𝑞∈𝔽[𝑥], which is a terminating CEF.

Acknowledgments

This work was supported by the Commission on Higher Education and the Thailand Research Fund RTA5180005 and by the Centre of Excellence In Mathematics, the Commission on Higher Education.