On the Sidon Ideal

#A91 INTEGERS 25 (2025)
ON THE SIDON IDEAL
And zej Nowik
Uni e si y o Gda´nsk, Ins i u e o Ma hema ics, Gda´nsk, Poland
[email p o ec ed]
Recei ed: 3/5/25, Accep ed: 9/23/25, Published: 11/5/25
Abs ac
We in es iga e he ideals o he na u al numbe s gene a ed by Sidon se s and weak
Sidon se s, p o ing ha hey a e, in ac , he same. We examine se e al p ope ies
o he esul ing ideal and place his ideal in he Kowi z Diag am o ideals.
1. In oduc ion
In ecen yea s, we ha e wi nessed he apid de elopmen o he heo y o ideals.
Many examples o ideals ha e al eady been conside ed; some o hem o igina e om
numbe heo y. Based on he well-known concep o a Sidon se , we de ine a new
ideal — namely he Sidon ideal — and de e mine i s place wi hin he amily o o he
known ideals. We deno e by N he se o all na u al numbe s wi h he numbe 0
adjoined. Fo in ini e se s A, B ⊆Nwe w i e A⊆∗Bi and only i A Bis ini e.
I A⊆Nis an in ini e se , hen A(n) deno es he n- h elemen o A. I A⊆N
and n∈N, hen A+n={a+n:a∈A}. I A, B a e nonemp y subse s o he
na u al numbe s, hen we de ine dis (A, B) = min{|a−b|:a∈A, b ∈B}. Also,
ecall he ollowing s anda d no a ion (see o example [3]): I A, B ⊆N hen de ine
A−B={a−b:a > b, a ∈A, b ∈B}and D(A) = A−A.
De ini ion 1. Suppose ha A⊆N. We de ine CenSym(A) by
{n∈A: he e exis s h∈N, h > 0 such ha n−h, n +h∈A}.
An ideal is a nonemp y amily I ⊂ P(N) closed unde aking subse s and ini e
unions, i.e.,
1. ∅∈I;
2. o all A∈ I and all B⊂A, we ha e B∈ I;
3. o all A, B ∈ I, we ha e A∪B∈ I.
DOI: 10.5281/zenodo.17535172
INTEGERS: 25 (2025) 2
An ideal is p ope i N∈ I, o , equi alen ly, I =P(N). Unless s a ed o he wise,
we assume all ideals a e p ope and we assume also ha all single ons belong o
he ideal. An ideal is dense i e e y in ini e subse o Ncon ains an in ini e subse
belonging o he ideal. An ideal Iis called a P-ideal i and only i , o each sequence
A0⊆∗A1⊆∗··· o se s om I, he e exis s A∗∈ I such ha An⊆A∗ o all n.
The ollowing ou de ini ions a e aken om [6].
De ini ion 2. The di e ence ideal Dwe de ine by
D={A⊆N: o all in ini e Z⊆N, D(Z)⊆ A}.
De ini ion 3. The D in ideal we de ine by
D in ={A⊆N: he e exis s n∈Nsuch ha o all Z∈[N]n, D(Z)⊆ A}.
De ini ion 4. An in ini e se A⊆Nis said o be a lacuna y se i and only i
limn→∞ A(n+ 1) −A(n) = ∞.
De ini ion 5. The Lacuna y ideal is he ideal gene a ed by lacuna y subse s and
we deno e i by Lac.
De ini ion 6. An in ini e se A⊆Nis a hin se i and only i limn→∞
A(n)
A(n+1) = 0.
De ini ion 7. An in ini e se A⊆Nis an almos hin se i and only i
lim sup
n→∞
A(n)
A(n+ 1) <1.
De ini ion 8. The ideal gene a ed by hin se s is called he hin se s ideal and we
deno e i by T. The ideal gene a ed by almos hin se s is called he almos hin
se s ideal and we deno e i by A.
De ini ion 9. Le Endeno e he amily o subse s o na u al numbe s which do
no con ain any a i hme ic sequence o leng h n. Le us de ine W as he amily o
subse s o na u al numbe s which do no con ain a i hme ic sequences o a bi a y
leng h n:
W = {A⊆N: he e exis s n > 0 such ha o all a∈N, > 0
he e exis s j < n such ha a+ ·j∈ A}.
This is an ideal and we call i he an de Wae den ideal.
2. The Sidon Ideal
Le us ecall he main ool o ou pape .
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De ini ion 10. A subse (sequence) A⊆Nis called a Sidon se i and only i
o n, m ∈A,n≤m, all sums n+ma e dis inc , i.e., o all n, m, k, l ∈A,
i n≤m, k ≤l, and n+m=k+l, hen n=k.
The ollowing de ini ion is aken om [5].
De ini ion 11. A subse (sequence) A⊆Nis called a weak Sidon se (o well-
sp ead) i o n, m ∈A,n<m, all sums n+ma e dis inc , i.e.,
o all n, m, k, l ∈A, i n < m, k < l, and n+m=k+l, hen n=k.
Recall he e ha he no ion o Sidon se s appea ed also in [3], page 6, and in [6]
unde he name D-spa se se s.
De ini ion 12. A se A⊆Nis D-spa se i o e e y k∈D(A) he e is only one
pai n, m ∈Awi h k=m−n.
No ice ha his no ion is equi alen o he so-called “Golomb ule ” (in ac ,
Golomb ule is a ini e e sion o D-spa se se s), see o example [2]. A simple
compu a ion shows ha he no ions o D-spa se se s and Sidon se s a e in ac
iden ical.
Le us de ine he main no ion o his pape .
De ini ion 13. The Sidon ideal, deno ed by Si, is he ideal gene a ed by he amily
o all Sidon se s, i.e.,
Si={A⊆N: he e exis s Sidon se s A1, . . . , Ansuch ha A⊆
n
[
j=1
Aj}.
I is also emp ing o de ine “a weak Sidon ideal” as an ideal gene a ed by all
weak Sidon se s, bu his ideal coincides wi h Si.
Theo em 1. E e y weak Sidon se can be pa i ioned in o wo Sidon se s.
P oo . I is well known ha , i Ais a weak Sidon se , hen A CenSym(A) is a Sidon
se . This is because he only equali y in he de ini ion o a Sidon se ha shows
ha a weak Sidon se Ais no a Sidon se is he equa ion n+m= 2 ·k,n < m,
whe e n=k, and his is he case when k∈CenSym(A). Now, assume ha Ais a
ini e weak Sidon se . De ine a sequence by se ing A0:= A;An+1 := CenSym(An).
This sequence is dec easing and since Ais a ini e se he e exis s n0such ha , o
all n>n0, we ha e An=∅( o a ini e Bwe canno ha e CenSym(B) = Bunless
B=∅). Le us de ine E(A) = S∞
k=0 A2k+1 A2k+2 and O(A) = S∞
k=0 A2k A2k+1.
O cou se, {E(A), O(A)}is a pa i ion o A.
We claim ha he se s E(A) and O(A) a e Sidon se s. To p o e he claim,
suppose by way o con adic ion ha he e exis nand h > 0 such ha n−h, n, n+
h∈E(A). Then n∈A2k0+1 A2k0+2 o some k0. Since n∈CenSym(A2k0), he e
INTEGERS: 25 (2025) 4
exis s h1>0 such ha n−h1, n, n +h1∈A2k0. I is clea ha , since Ais a weak
Sidon se , we ha e h=h1. Hence n−h, n +h∈A2k0. Now, suppose ha we ha e
n−h, n +h∈A2k0+1. Then we would ha e n, n −h, n +h∈A2k0+1 and his would
imply ha n∈CenSym(A2k0+1), so n∈A2k0+2, which would be impossible. Thus,
ei he n−h∈ A2k0+1 o n+h∈ A2k0+1. Suppose o example ha n−h∈ A2k0+1.
Then n−h∈A2k0 A2k0+1, so n−h∈O(A) which is a con adic ion. This p o es
he claim.
Thus, we ha e p o ed ha each ini e weak Sidon se can be pa i ioned in o wo
Sidon se s. Using a s anda d compac ness a gumen , we conclude ha any in ini e
weak Sidon se can be pa i ioned in o wo Sidon se s.
Co olla y 1. The ideal gene a ed by weak Sidon se s is he same as he ideal
gene a ed by Sidon se s.
No e ha he collec ion So all Sidon se s A⊆Nis a pe ec subse o he
Can o space 2N.
Co olla y 2. The Sidon ideal is an Fσideal.
P oo . Obse e ha he unc ion ∪n: (2ω)n→2ω, de ined by
∪n(A1, . . . , An) = Sn
j=1 Aj, is con inuous, and Si=S∞
n=1 ∪n[Sn].
3. The Place o he Sidon Ideal in he Kowi z Diag am
I is p o ed in [3] ha (see P oposi ion 4.3) o each in ini e A⊆N he e exis s an
in ini e D-spa se B⊆A. This shows ha he Sidon ideal is dense. Mo eo e , he
au ho p o es (see P oposi ion 4.3 (1)) ha i Ais D-spa se and n∈N hen A+n
is in he di e ence ideal. In pa icula , his shows ha he Sidon ideal is con ained
in he di e ence ideal.
Le us p o e a li le mo e.
Theo em 2. The Sidon ideal is con ained in he lacuna y ideal.
P oo . Suppose ha S⊆Nis a Sidon se , and le M > 0. Fo any m≤M he e
exis s a mos one pai {S(n), S(n+ 1)}such ha S(n+ 1) −S(n) = m. I he e
exis s such a pai , de ine nm=n. I he e is no such pai , de ine nm= 0. De ine
N= max{nm:m≤M}+ 1. Then, o n>N we ha e S(n+ 1) −S(n)> M, so S
is a lacuna y se .
I is s aigh o wa d ha he Sidon ideal is included in he Van de Wae den ideal.
This is because any Sidon se canno con ain a 3-elemen inc easing a i hme ic
sequence. Recall he ollowing esul s conce ning he exis ence o so-called “ a ”
Sidon se .
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Theo em 3. The e exis s an in ini e Sidon se Asuch ha
1. |[1, n]∩A|> c 3
√n o some c > 0(see [7]);
2. |[1, n]∩A|> c 3
pnlog(n)(see [1]);
3. |[1, n]∩A|> n√2−1−o(1) (see [8]).
E en he i s example o Theo em 3 su ices o p o e he ollowing p oposi ion,
which is a cons uc ion o a special Sidon se .
P oposi ion 1. The e exis s a Sidon se Awhich is no in he ideal A.
P oo . Le us begin wi h he ollowing (p obably well known) asymp o ic beha io
o any almos hin se . Suppose ha A⊆Nis an almos hin se . Then he e a e
C, D > 0 such ha |[0, M)∩A| ≤ Clog M+D. Le us ske ch he p oo . The e exis
N0>0 and α > 1 such ha A(n)
A(n+1) <1
α o n≥N0, so A(N0)·αn≤A(N0+n).
A simple compu a ion gi es us he es ima ion |(A(N0), M)∩A| ≤ logα(M); hence,
he e exis C, D > 0 such ha |[0, M)∩A| ≤ Clog M+D. Ob iously, he same
es ima ion holds o any elemen o he ideal A. By [7] he e exis s a Sidon se A
such ha |[1, M]∩A|> C1
3
√M o some C1>0. I his se we e an almos hin se
hen we would ha e C1
3
√M≤Clog(M) + D o any M, which is impossible.
Le us s a wi h an easy lemma.
Lemma 1. Suppose ha A⊆[1,∞)∩Nis a se such ha
2·A(n)≤A(n+ 1).(1)
Then Ais a Sidon se .
P oo . Suppose ha A(n) + A(m) = A(k) + A(l) o some n≤mand k≤l. I
l < m hen we would ha e A(k) + A(l)≤2A(l)≤A(l+ 1) ≤A(m)< A(m) + A(n),
which is impossible. Hence m=land he e o e n=k.
Theo em 4. E e y Ase is in he ideal Si.
P oo . I su ices o obse e ha since any almos hin se sa is ies he es ima e
A(N0)·αn≤A(N0+n), we can easily di ide i in o a ini e amoun o se s which
ul ills (1).
I is s aigh o wa d o see ha he Sidon ideal is no a P-ideal. We can use he
same example o se s (Ak={n! + k:n∈N}) om he Lemma 1.2.8 om [4] since
he se s Akbelong o Si. The Figu e 1 is a pa o he Kowi z Diag am (see [6],
page 20) wi h he new ideal Siin he app op ia e place.
We show ha all inclusions in he Figu e 1 conce ning he Sidon ideal a e i e-
e sible. By i ue o Example 1 i su ices o show he ollowing heo em.

INTEGERS: 25 (2025) 6
T
A
D in SiW
Lac
Figu e 1: A pa o he Kowi z diag am wi h he Sidon ideal
Theo em 5. The e exis s a se A∈W∩Lac ∩D in which is no in he ideal Si.
De ini ion 14. A se A⊆Nis said o be 3-a ihme ic ee i he e a e no a, b, c ∈A,
whe e a<b<c, such ha 2 ·b=a+c.
De ini ion 15. A se A⊆Nis said o be sum- ee i he e a e no a, b ∈A, whe e
a < b, such ha a+b∈A.
No e ha his de ini ion di e s om he s anda d de ini ion o a sum- ee se in
Schu ’s heo em, because we assume ha a<bno a≤b. No ice ha i a se A⊆N
is 3-a i hme ic ee hen A∈W. Also, i a se A⊆Nis sum- ee, hen A∈ D in —
his is a s anda d a gumen which mimics he a gumen om [3] ha a D-spa se
se is in he ideal D(see P oposi ion 4.3 (1) in [3]). Suppose ha n, C, M, K a e
na u al numbe s. De ine
Block(n, C, M, K) = {Ml+C·2k:k= 0,1, . . . , n −1; l= 1,2, . . . , K}.
A simple compu a ion shows ha i we assume ha M > C ·2n hen
Block(n, C, M, K) is bo h 3-a i hme ic and sum- ee. Le us o mula e he ollowing
lemma.
Lemma 2. Fo any sequence (Nk)o na u al numbe s and o any sequence (Ak)o
ini e se s, each o which a e bo h 3-a i hme ic and sum- ee se s o na u al numbe s,
we can ind a sequence (mk)o na u al numbe s such ha
1. he se A∗=S∞
k=1 Ak+mkis bo h 3-a i hme ic and sum- ee;
2. o all k, dis (Ak+mk, Ak+1 +mk+1)≥Nk.
P oo . The p oo is a s aigh o wa d induc i e cons uc ion.
Lemma 3. The se Block(n, C, C ·2n+ 1,(n−1) ·n
2+ 1) is no a sum o n−1
Sidon se s.
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P oo . By way o con adic ion, suppose ha Block(n, C, C·2n+1,(n−1)·n
2+1) =
Sn−1
j=1 Sj, whe e Sja e Sidon se s. Fo each l= 1, . . . (n−1) ·n
2+ 1 by he
pigeonhole p inciple we can ind k1(l), k2(l)∈ {0, . . . , n −1},k1(l)< k2(l), and
j(l)∈ {1, . . . , n −1}such ha Ml+C·2k1(l), Ml+C·2k2(l)∈Sj(l). Since he e
a e a mos (n−1) ·n
2 iple s (k1, k2, j)∈ {0, . . . , n −1}2×{1, . . . , n −1}, such
ha k1< k2, again by he pigeonhole p inciple we can ind l1< l2such ha
(k1(l1), k2(l1), j(l1)) = (k1(l2), k2(l2), j(l2)). Le ˆ
k1=k1(l1) = k1(l2), ˆ
k2=k2(l1) =
k2(l2), and ˆ
j=j(l1) = j(l2). Then we ha e Ml1+C·2ˆ
k1, Ml1+C·2ˆ
k2, Ml2+C·
2ˆ
k1, Ml2+C·2ˆ
k2∈Sˆ
jand his is impossible since Sˆ
jis a Sidon se .
P oo o Theo em 5. Now apply Lemma 2 o he sequences Nn=nand
An=Block(n, n, n ·2n+ 1,(n−1) ·n
2+ 1)
and we ob ain a sui able sequence (mn) and de ine a se
A∗=[
n=2
Block(n, n, n ·2n+ 1,(n−1) ·n
2+ 1) + mn.
By Lemma 2 his se is bo h 3-a i hme ic and sum- ee. Hence A∗∈W and
A∗∈ D in . I is easy o see ha also A∗∈ Lac. By Lemma 3, A∗is no a union o
ini ely many Sidon se s; he e o e, A∗∈ Si.
Acknowledgemen . I would like o exp ess my g a i ude o P o esso K zysz o
Kowi z and P o esso Jacek T yba o inspi ing and ui ul discussions.
Re e ences
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