ON SETS OF INTEGERS WHICH CONTAIN NO THREE TERMS IN GEOMETRIC PROGRESSION

Transcription

1 ON SETS OF INTEGERS WHICH CONTAIN NO THREE TERMS IN GEOMETRIC PROGRESSION NATHAN MCNEW Abstract The roblem of looing for subsets of the natural numbers which contain no 3-term arithmetic rogressions has a rich history Roth s theorem famously shows that any such subset cannot have ositive uer density In contrast, Ranin in 960 suggested looing at subsets without three-term geometric rogressions, and constructed such a subset with density about 079 More recently, several authors have found uer bounds for the uer density of such sets We significantly imrove uon these bounds, and demonstrate a method of constructing sets with a greater uer density than Ranin s set This construction is otimal in the sense that our method gives a way of effectively comuting the greatest ossible uer density of a geometric-rogressionfree set We also show that geometric rogressions in Z/nZ behave more lie Roth s theorem in that one cannot tae any fixed ositive roortion of the integers modulo a sufficiently large value of n while avoiding geometric rogressions Bacground Let A be a subset of the ositive integers A three-term arithmetic rogression in A is a rogression a, a + b, a + 2b A 3 with b > 0, or equivalently, a solution to the equation a + c = 2b where a, b and c are distinct elements of A We say that A is free of arithmetic rogressions if it contains no such rogressions For any subset A of the ositive integers we denote by da its asymtotic density if it exists and its uer density by da In 952 Roth roved the following famous theorem [6] Theorem Roth If A is a subset of the ositive integers with da > 0 then A contains a 3-term arithmetic rogression In articular, Roth showed that for any fixed α > 0 and sufficiently large N, any subset of the integers {,, N} of size at least αn contains a 3-term arithmetic rogression We can also view Roth s result as a statement about arithmetic rogressions with 3 distinct elements in the grou of integers modulo N Namely, if we denote by DZ/NZ the size of the largest subset of Z/N Z free of arithmetic rogressions, Roth s argument can be used to show [7] DZ/NZ = O N log log N This result has been imroved several times, see [9], [4], [9] with the current best result, due to Sanders [8] see also [3], Nlog log N 5 DZ/NZ = O log N Roth s theorem has since been generalized by Szemerédi to rogressions of arbitrary length Arithmetic-rogression-free sets have also been studied in the context of arbitrary abelian grous Meshulam [] generalized Roth s theorem to finite abelian grous of odd order, and recently Lev [0] has extended Meshulam s ideas to arbitrary finite abelian grous, a result which will be needed later Let DG be the size of the largest subset 200 Mathematics Subject Classification B05, B75, Y60, 05D0

2 of the finite grou G free of 3-term arithmetic rogressions If 2 haens to divide G, it is ossible to find examles of arithmetic rogressions with reeated terms by looing at elements of order 2, so we insist that our arithmetic rogressions consist of 3 distinct elements We also denote by cg the number of comonents of the abelian grou G when written in the invariant factor decomosition G = Z/ Z Z/ 2 Z Z/ t Z where 2 t, and write 2G for the grou of doubles {g + g : g G} Theorem 2 Lev For any abelian grou G, DG satisfies DG 2 G c2g Combining Lev s theorem with Sanders bound as in [, Corollary 3] one obtains Corollary 3 For any abelian grou G for which cg = c2g DG G log log G 5 log G 2 In a comletely analogous manner, one can consider geometric rogressions of integers of the form a, a, a 2 with Q and > equivalently, a solution to the equation ac = b 2 with distinct integers a, b, c and see sets of integers which are free of such rogressions It is somewhat surrising just how different the results are in this case We can immediately see, for examle, that the square-free integers, a set with asymtotic density 6 06, is free of geometric rogressions π 2 Unlie the difference of two terms in an arithmetic rogression, the ratio between successive terms of a geometric rogression of integers need not be an integer For examle, the rogression 4, 6, 9 is a geometric rogression with common ratio 3 While most of 2 the existing literature on this roblem is concerned only with the rational-ratio case, we will also consider the roblem restricted to integral ratios in the results that follow The roblem of finding sets of integers free of geometric rogressions was first considered by Ranin [4] who showed Theorem 4 There exists a set, G 3, free of geometric rogressions with asymtotic density dg 3 = ζ3 i ζ2 i> Here ζ is the Riemann zeta function ζ2 3 i Since then, several aers have further investigated the roblem of finding the largest ossible sets which are geometric-rogression-free We will use the notation α = su{ da : A N is free of geometric rogressions} α = su{da : A N is free of geometric rogressions and da exists} for sets free of geometric rogressions with rational ratio and β = su{ da : A N is free of integral ratio geometric rogressions} β = su{da : A N is free of integral ratio geometric rogressions and da exists} for the corresonding values when we restrict the roblem to integral ratios Clearly α α β and α β Ranin s construction remains the best lower bound for α, and was, rior to this wor, also the best lower bound for both α and β, though Beiglböc, Bergelson, Hindman, and Strauss [] had shown that β 075 More rogress had been made on the uer bounds for α Riddell [5] showed that α 6/7 This bound was reroved by Beiglböc, et al [], who were unaware of Riddell s result Their result was an imrovement of a bound obtained by Brown and Gordon [5] who, also unaware 2

3 of Riddell s result, had shown that α < Nathanson and O Bryant [3] combined these methods to show that α < While the existing literature on geometric-rogression-free sets has wored in greater generality, considering geometric rogressions of length 3, we have chosen in this aer to focus on rogressions of length 3 While most of the methods and constructions in this aer generalize to arbitrary, they don t, in general, aear to lead to a closed form exression in terms of We loo first at the roblem of finding geometric rogressions of the residues modulo n in Section 2 and find that for large enough n any ositive roortion of such residues will contain a geometric rogression Section 3 considers sets of integers free of rogressions with integral ratios and shows that one can construct such sets with substantially higher uer density than Ranin s set Our main results are in Section 4, where we demonstrate an algorithm for effectively comuting the value of α We use this method to significantly imrove several of the best nown bounds for the constants defined above Namely we show that and that < α < , < β < , < β Finally, in Section 6 we investigate bounds for sets which have an asymtotic density 2 Geometric Progressions in Z/nZ Ranin showed, in contrast to the arithmetic rogression case, that one can tae sets of integers with ositive density even the majority of integers and still avoid geometric rogressions In light of Ranin s result, it is somewhat surrising that this does not remain true when we loo at the integers modulo n We need, first, a roosition and a corollary of Lev s theorem for unit grous Z/nZ Proosition 2 Given any subgrou H of an abelian grou G DG G DH H Proof If A G is free of arithmetic rogressions, then for every coset xh the set A xh is free of arithmetic rogressions, and by the igeonhole rincile there exists at least one coset with A xh = x A H A H G Thus, since x A H is a subset of H free of arithmetic rogressions, A G x A H H G DH H Corollary 22 For any integer n, the grou Z/nZ of units modulo n satisfies DZ/nZ ϕnlog log n5 log n 2 3

4 Proof Let G = Z/nZ so that G = ϕn Write G = Z/2Z Z/2Z H where H is free of coies of Z/2Z in its invariant factor decomosition, and hence ch = c2h Let n = 2 2 ωn ωn be the rime factorization of n Since Z/nZ = Z/ Z Z/ 2 2 Z Z/ ωn ωn Z and each grou Z/ j j Z contains at most one coy of Z/2Z when j 2 and at most two if j = 2, we find that G has at most ωn + coies of Z/2Z Thus the subgrou ϕn H has size at least Thus, using Corollary 3 and Proosition 2, 2 ωn+ DG G DH H G H log log H 5 H log H 2 ϕnlog log n5 log ϕn 2 ωn+ 2 Using the facts [8, Theorem 323 and Section 220] that lim inf ωn = O, we have log n log log n ϕn log log n n = e γ and log ϕn/2 ωn+ log = log n + O n e γ log log n log n log log n + o ωn + log2 So DG ϕnlog log n5 log n 2 Theorem 23 Let EZ/nZ denote the size of the largest ossible subset of the residues modulo n which does not contain a 3-term geometric rogression Then EZ/nZ nlog log n5 log n /2 Proof For each d n let R d denote the set of m Z/nZ such that m, n = d So, for examle, when d =, R = Z/nZ Note that R d can be viewed as dz/ n d Z in the sense that each element of R d is uniquely reresentable as d times a residue corime to n d Thus R d = ϕ n d Furthermore, any arithmetic rogression, a, ab, ab 2, written multilicatively in the grou Z/ n d Z corresonds to the geometric rogression da, dab, dab 2 contained in the set of residues R d So any geometric-rogression-free subset of R d cannot be larger 4

5 than DZ/ n d Z Because the R d artition Z/nZ, we see that EZ/nZ d n DZ/dZ = DZ/dZ + d n d< d n n d n d n d< n < d n n + d n d n DZ/dZ ϕdlog log d 5 log d 2 log log n 5 n + log n ϕd 2 2 d n n /2+ɛ nlog log n5 nlog log n5 + log n 2 log n 2 Where we used the fact that the number of divisors of n is On ɛ for every ɛ > 0 Bounding the exonent on log n in this theorem is comlicated by the fact that the size, λn, of the largest cyclic subgrou of the unit grou Z/nZ can occasionally be very small In general, however, this is not the case In fact, we now [6, Lemma 2] that there exists a set S of integers with asymtotic density such that for all n in S, λn = n/log n log log log n+o Let n S and let H be a cyclic subgrou of Z/nZ of size λn Then, using Sanders bound and Proosition 2 DZ/nZ Z/nZ DH H ϕn H log log H 5 H log H We now [7, Lemma 2] that for each d n, so for n S, and d n, λd = ϕnlog logn/log nlog log log n+o 5 logn/log n log log log n+o ϕnlog log n5 log n λd d λn n, 2 d If we assume d > n then the above log n log log log n+o argument shows that DZ/dZ ϕdlog log d5 Using this inequality in lace of log d Corollary 22 in line 2 of the roof above, we have Theorem 24 For n S, EZ/nZ nlog log n5 log n 3 Geometric Progressions with Integral Ratio As mentioned before, it is ossible to consider geometric rogressions of two different tyes, deending on whether or not the ratio common to the rogression is an integer One would exect that restricting to the integral case should allow us to construct sets with larger asymtotic density, since there are fewer restrictions on our set While Ranin s set G 3 is constructed to avoid rational geometric rogressions, no integers can be added to it without introducing an integer geometric rogression Nevertheless, we can construct sets 5

6 with substantially higher uer density The following construction extends the method described in [, Theorem 32] Theorem 3 We have the lower bound β > Proof Note that for any N, the set N, N] is free of geometric rogressions with integral 4 ratio since for any n N, N] and r 2, we have nr 2 > N This observation, combined 4 with the argument described below, could be used to construct a set with uer density 3 We can, however, do better 4 Rather than just using the range N, N], we note that the set N, ] N N, N] also has 4 the roerty of being free of geometric rogressions since for any n N, ] N 9 8, the integer 2n lies in the omitted interval N, ] N 8 4, hence n is not art of a geometric rogression with common ratio 2, and 9n > N, meaning n cannot be art of a rogression of common ratio greater than or equal to 3 One can further chec that the set N S N = 48, N ] N 45 40, N ] N 36 32, N ] N 27 24, N ] N 2 9, N ] ] N 8 4, N has this roerty, and for sufficiently large N will contain at least 3523N > N integers less than N We can continue this rocess, adding smaller and smaller intervals to S N indefinitely, and create sets with marginally greater density However, we can t tae another such interval until N/2208 Now, fix N = N, and let N 2 = 48 2 N The set S N S N2 will also be free of geometric rogressions with integral ratio, since if n, nr S N then r < 48 so nr 2 < 48N = N 2 /48, and so nr 2 S N2 Similarly, if nr, nr 2 S N2, we again have r < 48 and so thus n S N In general, set n > nr 48 > N = N N i = 482 Ni 2 N Then any geometric rogression with two elements contained in S Ni will not have a third element in the union of the S Nj Furthermore, if n S Ni and nr S Nj for some j > i, then r < 48N j N i < 48N j N no geometric rogression can have terms in 3 distinct S Ni, so nr 2 < 48N 2 j N = N j+ 48 and so nr 2 S N for any j As a result, Thus, letting S be the union of all such S Ni, we find that ds > , and the entire set S is free of geometric rogressions with integral ratio Kevin Ford oints out that a slightly higher uer density can be achieved by letting N S N = 48, N ] ] N { N 2 4, N 9 < n N } : 3 n, 4 n, 5 n 8 By a similar argument this set is also free of geometric rogressions with integral ratio In this case any geometric rogression which tries to bridge the ga between the intervals will have a term divisible by 3, 4 or 5 in the middle interval Setting S N = S N S N 48 we see that S N is free of geometric rogressions since each of its comonents is, and any geometric rogression with terms in both comonents would need to have ratio greater than 48 to jum the ga Constructing S using widely searated coies of S N as above we find that ds = > 0884, giving us the following stronger bound 6

7 Theorem 32 Ford We have the better lower bound β > We are also able to construct sets free of geometric rogressions with integral ratio with a slightly higher asymtotic density than the set generated by the greedy algorithm, described by Ranin For convenience we recall the roof of Theorem 4, which constructs Ranin s set, G 3 Proof Note that if a, a, a 2 is a geometric rogression and we denote by v a the -adic valuation of a, then v a, v a, v a 2 forms an arithmetic rogression which is non-trivial if v 0 Thus, any set of integers, all of whose rime factors occur with exonent contained in a set A free of arithmetic rogressions, will be free of geometric rogressions [5, Theorem ] Tae A = A 3 = {0,, 3, 4, 9, } to be the set of nonnegative integers which do not have a digit two in their ternary exansions This is the set obtained by choosing integers free of arithmetic rogressions using a greedy algorithm Now, letting G 3 = {n N : for all rimes, v n A 3}, we obtain a set free of geometric rogressions Note that this set is also the set obtained using a greedy algorithm to choose ositive integers free of geometric rogressions, either of integer or rational ratio [5] The density of this set, G 3, can be found using an Euler roduct reminiscent of the Euler roduct for the density of squarefree numbers For a fixed rime the asymtotic density of those integers divisible by an accetable ower of the rime is given by = i = + 2 i>0 i>0 + 3i + Then, by the Chinese remainder theorem, we can comute the density of G 3 as a roduct over all rimes, dg 3 = + = ζ3 i 2 3i ζ2 ζ2 3 i i>0 i> So, for examle, considering just the rimes 2, 3 and 5, Ranin s construction would include numbers of the form 2 3 5, where is an integer divisible only by rimes larger than 5, for each rime > 5, v A 3 If, rather, we were to exclude integers of this form, we would be able to include numbers of the following forms: , , , , , and Each of these new numbers we ve included will force us to exclude numbers with these rimes to higher owers, but in the end calculating these inclusion/exclusions with a comuter we find that we gain about asymtotically in the trade, roducing a new set with asymtotic density This roves a new lower bound for the constant β, defined in Section Theorem 33 We have the lower bound β > This can be further imroved by incororating more rimes and exclusions Note, however, that this larger set does include rogressions with rational ratios For examle, rogressions of the form 2 3 5, , are included even though they form a rogression with common ratio i

8 4 Uer Bounds Uer bounds for the densities of sets of integers free of geometric rogressions have been studied in several aers, with the current best bound being that of Nathanson and O Bryant, that the uer density of any geometric-rogression-free set is at most Riddell [5] gives the uer bound 08339, but states that The details are too lengthy to be included here In this section we imrove this bound to We consider first the roblem of avoiding geometric rogressions with ratios involving only a finite set of rimes, in articular the rimes smaller than some bound s Denote by g s N the cardinality of the largest subset of the integers {,, N} which is free of 3-term rational-ratio geometric rogressions which have common ratio involving only the rimes less than or equal to s We will see in Theorem 42 that for any s the limit g lim sn N exists and furthermore is equal to the suremum of the uer densities of N sets of ositive integers which avoid such rogressions We will denote this value by α s = su{ da : A N is free of s-smooth rational geometric rogressions} The simlest case, s = 2, in which all of the ratios involved are integral, was recently studied along with sets avoiding geometric rogressions whose ratios involve any one, single, integer by Nathanson and O Bryant [2] They show that α 2 is an irrational number aroximately with error less than Before roving the result for general s, we consider the secific case of just the rimes 2 and 3, which is illustrative of the general behavior when multile rimes are involved g Claim 4 The limit lim 3 N N exists and is equal to α N 3 Furthermore, this quantity is bounded by < α 3 < Proof Fix N > 0 and consider the largest subset of the integers {,, N} free of geometric rogressions which have a common ratio involving only the rimes 2 and 3 Denote by S 3 the set of 3-smooth numbers numbers whose only rime divisors are 2 and 3 at most Note first that any geometric-rogression-free subset of S 3 4 = {, 2, 3, 4} must exclude at least one integer from this set, and hence for any integer b N such 4 that b, 6 = our set must exclude at least one of the integers b, 2b, 3b, 4b The single one excluded cannot be 3b Since these numbers are distinct for different values of b, and = φ6 3 of the integers less than N are corime to 6, we must exclude a total of at 6 4 least N O integers If we now consider S 3 9 = {, 2, 3, 4, 6, 8, 9}, we find that this set contains the 4 rogressions, 2, 4, 2, 4, 8,, 3, 9 and 4, 6, 9 which cannot all be recluded by removing any single number However, removing the two integers 2 and 9 suffices This means that for each b N, b, 6 =, we must exclude at least two of the integers from the set 9 {b, 2b, 3b, 4b, 6b, 8b, 9b}, and moreover these sets are disjoint, not only from each other, but also simly extend the sets constructed from S 3 4 above Thus each b N with 9 b, 6 = corresonds to an additional excluded integer, meaning we must now exclude at least N + N O integers In general, for each we can find a not necessarily unique subset T 3 S 3 which is as large as ossible while avoiding geometric rogressions For each value of at which the number of necessary exclusions, m = S 3 3 T increases a new exclusion N is required there are an additional integers which must be excluded from our set 3 One can chec comutationally that the first few values of which require an additional exclusion are given in the following table 8

9 Table # of integers # of integers # of integers excluded from S 3 excluded from S 3 excluded from S Taing all these exclusions into account, we find that we ve excluded N O > , we have that g3 N < N for N numbers Since 3 sufficiently large Note that the rocess described above is also constructive For a fixed integer N we can tae, for each b N, b, 6 =, the set of integers b T 3 fb where fb = { N b b > N Otherwise and not include any rogressions involving the rimes 2, and 3 Taing the union over all such b, the set we construct will differ in size from our uer bound only by the trailing terms in the series of 3-smooth numbers greater than 5832 which we have not yet taen into account Since n>5832 n is 3-smooth 3n < , we have that g 3 N > N As we tae into account more 3-smooth numbers the trailing recirocal sum of these numbers above will become arbitrarily small, so we see g that the limit lim 3 N N exists N We can now go one ste further and use this idea to construct a set of ositive integers which has this uer density As in Theorem 3, we can iece together widely searated sets of integers each constructed by the method just described The resulting set can have uer density arbitrarily close to the value above, while avoiding geometric rogressions involving the rimes 2 and 3 This construction therefore gives us a lower bound for α 3, the suremum of the uer densities of all such sets Since α 3 must also satisfy the same uer bound as g 3N, we N g can conclude that in fact α 3 = lim 3 N N N 9

10 There is nothing articular about the rimes 2 and 3 in this argument In general, write the sequence of s-smooth numbers = n < n 2 < in increasing order For each j let T n s j denote a subset of S n s j = {n, n 2,, n j } which is as large as ossible though not necessarily unique while avoiding triles in geometric rogression, and let m nj = S n s j T n s j Note that the m nj are nondecreasing Let I s be the set of numbers j with m nj > m nj those j for which S n s j requires an additional exclusion For s = 3 these are the numbers aearing in Table We have the following result Theorem 42 For each integer s 2, α s = s n j j I s Since any geometric-rogression-free subset of the integers must be free of ratios that involve only the rimes 2 and 3, we see that the uer bound above, , is also an uer bound for α Already this value is better than the bounds given before in the literature, but we can imrove this result further If we consider now the rimes 2, 3 and 5, we see that the roof above goes through in exactly the same way, requiring additional exclusions at each of the integers which are the first 36 exclusions required This list gives us the bounds < α 5 < The difficulty in ushing this method further is the amount of comutation required to find the largest geometric-rogression-free subset of S 5 the 5-smooth numbers u to for increasingly larger For examle, showing that an additional exclusion is required at 576 the next 5-smooth number after 540 would require showing that there are no geometric-rogression-free subsets of size 36 among the 70 5-smooth numbers u to 576 Even though comutational limitations revent us from finding the exact values where additional exclusions are necessary ast 540 in S 5 we can still use some of the comutational wor we did to estimate α 3 to further imrove the uer bound on α 5 Taing I 3 the set of values where an additional exclusion was required to avoid 3-smooth rationalratio geometric rogressions among the 3-smooth integers and multilying its elements by each ower of 5, we obtain a subset of the 5-smooth numbers, J 5,3 = i 0 5 i I 3 3 J 5,3 is the set of values at which an additional exclusion is required to avoid 3-smooth rational-ratio rogressions among the 5-smooth integers Since avoiding 5-smooth rogressions will only require more exclusions, we now that the ith smallest entry of J 5,3 is greater than or equal to the ith entry of I 5, and thus can be used as an uer bound for this value So, multilying 4, 9, 6, 8, , the numbers from Table the first 36 elements of I 3 by, 5, 25, 25 and reordering we obtain the first few entries of J 5,3 :

11 Note that each term in this list 4 is greater than or equal to the corresonding term in 3 Looing at the 37th entry of this table we see that we will require an additional exclusion by the time we reach 800 So, taing all of these exclusions into account first the 36 exclusions from 3, and then those starting at 800 from 4 we can decrease our bound by an additional , so α 5 < Alying this rocess again for the rimes 2, 3, 5 and 7, where we comute that exclusions must be made at and incororating the exclusions calculated for both the 2, 3 and 2, 3, 5 cases, as described above, we obtain the bound α 7 < Again, this is an uer bound for the uer density of a set of integers avoiding all geometric rogressions which roves the following Theorem 43 We have α < Since this uer bound is lower than the uer density of the set we constructed for the integer-ratio roblem in Theorem 3, we see as one might have exected that these two roblems, considering integer and rational ratios, are in fact different One can carry through an analogous argument considering only rogressions with integral ratios, in which case we find looing at 3-smooth integer rogressions that we must mae exclusions at yielding an uer bound of If we combine this argument, as in the rational-ratio case above, with the necessary exclusions for 5-smooth rogressions, we can comute a new uer bound for β which is less than 000 above the lower bound of the set constructed in section 3 Theorem 44 We have < β < We can also use the set we constructed in the roof of Claim 4, which had high uer density while avoiding geometric rogressions involving the rimes 2 and 3, to construct sets free of any rational-ratio rogression which have a higher uer density than Ranin s set Theorem 45 There exist geometric-rogression-free sets with uer density greater than , so < α < Proof Recall that Ranin s construction consisted of integers with exonents on rimes contained in the set A 3, the greedily chosen set free of arithmetic rogressions To construct a set with greater uer density, we start with the set described in Claim 4, denoted here by H 3, which is free of geometric rogressions involving the rimes 2 and 3 Recall that dh 3 > We now remove from H 3 those integers which have a rime greater than 3 with exonent not contained in A 3 Symbolically we construct H 3 = {n H 3 : 5, v n A 3} Essentially, rather than taing all integers, b, corime to 6 in the argument above, we use Ranin s construction to choose integers corime to 6 which do not themselves contain 5

12 any geometric rogressions We then have found a more efficient way in regard to uer density of choosing exonents for the rimes 2 and 3 than Ranin s method To find the uer density of this set, we recall the Euler roduct of the density of Ranin s set, ζ2 i>0 ζ3 i ζ2 3 i = Now, fix an integer N and consider, for examle, the interval [N/6, N/8 Since 4 required one exclusion and 6 did not yet require an additional exclusion, we could tae for each b [N/6, N/8 with b, 6 = four of the five integers b, 2b, 3b, 4b, 6b without introducing a 3-smooth geometric rogression, a total contribution of 4 N N + O integers less than N We now add the further restriction that our integers b must be in the set G 3 In order to avoid rogressions involving other rimes as well as being corime to 6 Such candidates for b have asymtotic density 3 5 i and so the contribution from the range [N/6, N/8 is now 4 N 3 4 N i + O 6 5 Doing this for every interval gives us a contribution of N n and so, dh 3 > n>5832 n is 3-smooth 5 Comuting α i 5 i > i + O The arguments given above show, not only how to comute good aroximations for each α s which we can use to bound α, but also that these values converge to α Theorem 5 In the limit, as a larger set of initial rimes is taen into account, lim s α s = α Proof The arguments of Theorems 45 and 42 show that α s i α α s >s and since this Euler roduct converges, we now that lim i = s The conclusion follows >s 2

13 Thus, since we have a method to comute each α s to any desired recision, we can also do so for the constant α by extending the methods described above We loo here at the comlexity of comuting α Theorem 52 For each number ɛ with 0 < ɛ <, the constant α can be comuted to within ɛ in time O log 2 ɛ ɛ Proof We need, first, to consider a sufficient number of rimes so that i > ɛ/2, >s where A 3 is the greedily chosen set of integers free of arithmetic rogressions used in Ranin s construction Using the inequality i > + = 2 >s >s >s > n = n n + = s 2 n 2 s +, n>s n>s we see that taing s > ɛ/2 = 2 ɛ suffices ɛ/2 ɛ Second, we need to comute α s to sufficient accuracy so that the trailing terms in our series of recirocals of s-smooth numbers is less than ɛ/2 Let N = log 2 Then i 2 = i 2 ɛ N 2πs i>n i>n s ɛ 2πs Then the error in aroximating α s by using the N + πs s-smooth integers BN s = {2 i 3 i2 i πs πs : 0 i N} is less than n < n n is s-smooth n B s N = s = s s s n is s-smooth v n>n s i>n i>n i < i s q q q s i 0 ɛ 2πs = ɛ 2 Now, using the N + πs smallest s-smooth integers, rather than those in BN s, will only mae the error smaller So it will suffice for our comutation to wor with the exclusions required among the first N + πs s-smooth integers In articular, we need to calculate for each j K the minimal number of exclusions required from the set of integers S s j We need to exclude at least one member of each 3-term geometric rogression contained in this set, an examle of a 3-hitting set roblem, which is a roblem nown to be NPcomlete In [20] Wahlström gives an algorithm for comuting a 3-hitting set for a set of 3

14 size n and any collection of 3-element subsets of it in time O 6538 n He also gives an algorithm that requires exonential sace but runs in time O 638 n Using this algorithm for each j K to comute the minimal number of exclusions will require time O j K 6538j = O 6538 K in total Substituting in the definitions of K and N, we see this taes time O 6538 log 2 ɛ 2π 2 ɛ ɛ + π 2 ɛ ɛ = O log 8π 2 ɛ ɛ ɛ π 2 ɛ ɛ or, using the crude inequality π 2 ɛ ɛ < 8π 2 ɛ ɛ < for sufficiently small ɛ, ɛ O log 2 ɛ ɛ Let n, n 2, n l be the exclusion oints returned by Wahlstrom s algorithm and set M = l n s i= i We see that M > α > M ɛ i 2 >s > M ɛ ɛ > M ɛ 2 2 and thus we have achieved the required recision in our estimate of α The same arguments aly for the constant β as well While it aears from our comutations so far that the argument of Theorem 32 is far more efficient at comuting lower bounds for β than the analogous argument to that described here for integer-ratios, it is not clear that the construction described there converges to β 6 Sets with Asymtotic Density All of the uer bounds given here and elsewhere in the literature are for the uer density of geometric-rogression-free sets, while the set Ranin constructed to roduce a lower bound has an asymtotic density It is quite ossible that the restricted collection of such sets which ossess an asymtotic density will have smaller densities We can rove that this is the case when avoiding rogressions involving only one rime In fact, we show this under the weaer hyothesis that such a set has logarithmic density Any set, S, with asymtotic density also has logarithmic density equal to its asymtotic one, ds = lim x log x s s S s x We show here that any set with logarithmic density and hence any set with asymtotic density free of geometric rogressions involving owers of 2 will have density strictly smaller than α , the suremum of the uer densities of all 2-smooth geometric-rogression-free sets mentioned at the end of Section 4 4

15 Theorem 6 The set, T = {n N : v 2 n A 3} where A 3 is the set free of arithmetic rogressions obtained by the greedy algorithm, has the largest density among all sets which are free of 2-smooth-geometric rogressions of length 3 and have a logarithmic density This set, T, has asymtotic density, dt = 2 2 i < α 2 Proof The inclusion of any integer, s, in a set, S, avoiding 2-smooth geometric rogressions only affects the ossible inclusion of terms of the form 2 i s later in the set Now, because s = i>0 we see that the contribution of any term, s, to the sum in the logarithmic density of S is greater than any ossible contribution from later terms in the rogression of multiles of s by owers of 2 The greedy set is thus seen to be otimal in this case: The contribution to the logarithmic density of any integer, s, outweighs any otential benefit which could be gained by excluding it The set T is the set obtained by the greedy algorithm in this case Each integer t is included in T if doing so does not create a 2-smooth geometric rogression with smaller integers already in T, which occurs if and only if the exonent of the greatest ower of 2 dividing t is in the set A 3 This argument is readily seen to generalize to sets avoiding geometric rogressions with ratios involving any single rime, or even to ratios comosed of any fixed set of rimes P with the roerty that P 2 In each case the greedy set is seen to be otimal among sets which have logarithmic asymtotic density and avoid geometric rogressions with such ratios 2 i s 7 Oen Questions This aer answers one of the questions in [3] by demonstrating a method of effectively comuting the value of α The question osed in that aer ass for the maximal uer density of sets avoiding rogressions of arbitrary length, Our constant, α, is defined in regards to rogressions of length 3, but the methods here easily generalize to rogressions of any length, Their second question, however, regarding the recise value of α remains oen We do not even now the answer to the following question Question 7 Is α strictly smaller than α? This seems almost certain to be true, esecially given the result of Section 6 In light of this result, and the comutations required for Theorem 33 we mae the stronger conjecture Conjecture 72 Ranin s set, G 3 has the largest ossible density among geometricrogression-free sets which have an asymtotic density, so α = dg 3 One can as the same question about β Question 73 Is β < β? Question 74 While we now from Theorem 33 that β > dg 3, and from Section 4 that β > α, do we have β > α? In [] and [2] Beiglböc, Bergelson, Hindman, and Strauss tae a more Ramseytheoretic view of the roblem They observe, for examle, that a van der Waerden tye theorem holds for geometric rogressions: For any -coloring of the natural numbers, there are arbitrarily long monochromatic integral ratio geometric rogressions One 5

16 of the questions in [] can be answered with the methods of their aer They as if there is a set A of ositive integers, free of 3-term rational-ratio geometric rogressions, such that A has ositive uer density and A contains arbitrarily long intervals Such a set can be constructed by alternating between long runs from Ranin s set of ositive density, long gas with no integers, and consecutive integers in an interval of the shae [x, x + x ] Their Lemma 33 imlies such an interval of consecutive integers has no 3-term rational-ratio geometric rogressions Modifying this slightly, the set can even be taen to have ositive lower density Thans are due to Carl Pomerance for these observations Here is a nice roblem from [] that remains unsolved Question 75 Must every infinite set of natural numbers with bounded gas between consecutive terms contain arbitrarily long geometric rogressions? Acnowledgments I would lie to than Kevin Ford for suggesting the roblem of geometric rogressions in Z/nZ and for allowing me to include his imrovement, Theorem 32 I would also lie to than my advisor, Carl Pomerance, for his suort and invaluable guidance during the develoment of this aer and the referee for many useful suggestions References M Beiglböc, V Bergelson, N Hindman, and D Strauss, Multilicative structures in additively large sets, J Combin Theory Ser A , no 7, MR f:0574 2, Some new results in multilicative and additive Ramsey theory, Trans Amer Math Soc , no 2, MR g: T F Bloom, A quantitative imrovement for Roth s theorem on arithmetic rogressions, arxiv rerint arxiv: J Bourgain, On triles in arithmetic rogression, Geom Funct Anal 9 999, no 5, MR h:32 5 B E Brown and D M Gordon, On sequences without geometric rogressions, Math Com , no 26, MR a:024 6 P Erdős, C Pomerance, and E Schmutz, Carmichael s lambda function, Acta Arith 58 99, no 4, MR g:093 7 J Friedlander, C Pomerance, and I Sharlinsi, Period of the ower generator and small values of Carmichael s function, Math Com , no 236, , Corrigendum in , G H Hardy and E M Wright, An introduction to the theory of numbers, Oxford University Press, D R Heath-Brown, Integer sets containing no arithmetic rogressions, J London Math Soc , no 3, MR g:005 0 V F Lev, Progression-free sets in finite abelian grous, J Number Theory , no, MR :023 R Meshulam, On subsets of finite abelian grous with no 3-term arithmetic rogressions, J Combin Theory Ser A 7 995, no, MR g: M B Nathanson and K O Bryant, Irrational numbers associated to sequences without geometric rogressions, arxiv rerint arxiv: , On sequences without geometric rogressions, Integers 3 203, Paer No A73, 5 MR R A Ranin, Sets of integers containing not more than a given number of terms in arithmetical rogression, Proc Roy Soc Edinburgh Sect A /96, /6 MR #95 5 J Riddell, Sets of integers containing no n terms in geometric rogression, Glasgow Math J 0 969, MR #677 6 K F Roth, Sur quelques ensembles d entiers, C R Acad Sci Paris , MR ,724d 6

17 7, On certain sets of integers, J London Math Soc , MR ,536g 8 T Sanders, On Roth s theorem on rogressions, Ann of Math , no, MR f:09 9 E Szemerédi, Integer sets containing no arithmetic rogressions, Acta Math Hungar , no -2, MR c:00 20 M Wahlström, Exact algorithms for finding minimum transversals in ran-3 hyergrahs, Journal of Algorithms , no 2, 07 2 Deartment of Mathematics, Dartmouth College, Hanover, NH USA address: nathangmcnewgr@dartmouthedu 7