Abstract: Bulletin of the London Mathematical SocietyVolume 53, Issue 6 p. 1916-1949 OBITUARYFree Access Alan Baker, FRS, 1939–2018 David Masser, Corresponding Author David Masser [email protected] Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, Basel, 4051 Switzerland [email protected]Search for more papers by this author David Masser, Corresponding Author David Masser [email protected] Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, Basel, 4051 Switzerland [email protected]Search for more papers by this author First published: 30 December 2021 https://doi.org/10.1112/blms.12553Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract Alan Baker, Fields Medallist, died on 4 February 2018 in Cambridge, England, after a severe stroke a few days earlier. In 1970 he was awarded the Fields Medal at the International Congress in Nice on the basis of his outstanding work on linear forms in logarithms and its consequences. Since then he received many honours, including the prestigious Adams Prize of Cambridge University, the election to the Royal Society (1973) and the Academia Europeae; and he was made an honorary fellow of University College London, a foreign fellow of the Indian Academy of Science, a foreign fellow of the National Academy of Sciences, India, an honorary member of the Hungarian Academy of Sciences, and a fellow of the American Mathematical Society. 1 Life and career Alan's paternal grandparents were known as Marks and Mathilda Backer; they were married in 1902 in Lithuania. The surname presumably changed when children arrived. His parents were Barnet and Bessie (née Sohn). Alan was born in London on 19 August 1939 into this Jewish family. His earliest memories of wartime England were of evacuation to Camberley, Surrey. After the war, the family moved to Forest Gate in East London, where he spent most of his early life. From a very early age, he showed signs of mathematical brilliance (see the comments later about 'brainbox') and was encouraged by his parents. Already his father (who had been at school with Jacob Bronowski) was very gifted in this direction, but did not have the opportunity to develop, and became a tailor instead. This may have explained Alan's clothes sense; he was always well-turned-out with quality suits and tasteful ties (which were, however, not always entirely appropriate to his later travelling, for example, on the beach at Nice after being awarded the Fields Medal, or climbing a hill in the Australian Bush among snakes — such episodes may have contributed to his later investing in a distinctive yellow sun hat). Alan's first education was at a Franciscan Convent (London E7) 1945–1949, followed by Godwin County Primary School 1949–1950 and Stratford Grammar School 1950–1958. He then went with a state scholarship to University College, London (he recalls that the staff included J. W. Archbold, L. S. Bosanquet, P. Du Val, T. Estermann, C. A. Rogers, and K. Roth), where he studied mathematics 1958–1961, obtaining a first-class honours BSc (Special) degree. He then moved to Trinity College, Cambridge (where he would soon be based for the rest of his life) to study from 1961 to 1964 with Harold Davenport, at the time one of the leading number theorists with many international connections, of whom Alan writes: 'An excellent mathematician from whom I learnt a great deal; but I tended to follow my own lines of research.' Still, Chris Morley recalls a loud mathematical conversation between them in the Trinity Parlour, and they did write a paper together (see below). It seems that in later life he enjoyed imitating Davenport's Lancashire (Accrington) accent (and the writer has carried on this worthy tradition of imitating one's supervisor). He obtained his PhD in 1965 and MA in 1966, by which time he had already been awarded a Prize Fellowship for 1964–1968, also at Trinity. During this period, which included a year 1964–1965 at University College, he took on John Coates as a PhD student, and recalls also sharing responsibilities with Davenport in supervising T. W. Cusick, M. N. Huxley, H. L. Montgomery, and R. W. K. Odoni. In 1966, he was appointed as Assistant Lecturer and in 1968 as Lecturer; in 1972 he was promoted to Reader in the Theory of Numbers, and finally in 1974 he was elected to a personal chair for Pure Mathematics, all at Cambridge. Apparently he liked to draw attention to the unusual chronological order of Fields Medal, then Fellow of the Royal Society, then Professor. During that period he supervised for a PhD also, in chronological order, the writer, Cameron Stewart, Yuval Flicker, Roger Heath-Brown, Richard Mason, Mark Coleman, and Ellyn Lee. At the time of writing, the Mathematical Genealogy Project lists 506 descendants. The writer recalls meeting Alan every two weeks or so in his college rooms in Whewell's Court opposite the Great Gate. Actually these rooms (fitting C. P. Snow's description 'not specially agreeable') had been occupied previously by G. H. Hardy (about which the great German mathematician David Hilbert — see later — indignantly wrote to the Master that Hardy was the best mathematician, not only in Trinity, but in England, and should therefore have the best rooms). One had to ascend a slightly low and narrow spiral staircase, and then knock on the thick wooden door. Often in College une porte peut en cacher une autre and probably he did indeed have a second door immediately behind; at any rate, he was sometimes a long time coming, and I supposed that he did not hear me through the two doors (also against the heavy traffic then in Trinity Street). On each visit, I felt obliged to knock or bang harder and yet harder, sometimes bringing with me a heavy book to protect my hands. As mentioned, Alan Baker was firmly based in Cambridge; it seems that college life there suited him — especially in the style of Trinity, whose society he enriched for many years in an unspectacular way, for example by being an interesting conversationalist (contrary to the impression given by some newspaper obituaries). He also threw after-seminar parties fuelled by particularly strong beer (on one occasion Frank Adams challenged others to climb around a table, underneath it, without the feet touching the ground, after he had himself demonstrated that it could be done). In the Combination Room, there was a bottle of Madeira kept especially for him. He was a reasonable ballroom dancer, enthusiastically participating in the College May Ball. He played regularly on the Trinity Bowling Green, using an unconventional throw which delivered the ball from waist level instead of lower down. Outside the College, he enjoyed playing the slot machines in pubs, or playing table-tennis and snooker at the Graduate Centre. Eva McLean (née Gordon) writes of one occasion: 'In 1975 a vicious rapist was terrorising the female population of Cambridge. One evening when I was leaving the Graduate Centre one of the porters who knew me well expressed great concern that I would be walking home alone. Not to worry, I assured him brightly, pointing to Alan who was going to escort me. The burly ex-policeman looked him up and down, all five foot six of him, grabbed his coat and joined us on our way.' She also draws attention to existential dangers of a different kind: 'Alan was once barricaded in his flat for several days and would not answer the door as a collector, all the way from America, kept coming back pleading, in vain, for a contribution to his sperm bank.' He had learnt to drive in America, and in Cambridge he bought a Rover car, apparently as an investment ('you understand', as he insisted), although Eva McLean soon explained to him that there were much better ways of investing (and he seems to have acted very well on this advice). He drove with enthusiasm despite colliding with a stationary fire engine on his maiden voyage, and despite being stopped for speeding, which caused him to arrive late in chairing a session in London. William Chen recalls that it was actually the inaugural lecture of a colleague, and that Alan was very eager to explain that he had not exceeded 90 miles per hour, so he was only speeding rather than driving recklessly. On one occasion, the car got badly stuck in a driveway; Alan, however, remained cool and concentrated and succeeded in extricating it without damage. Later on the enthusiasm waned, and a rusting hulk with flat tyres had to be removed from New Court. After that, he reverted to his old mode of transport, which was not by bicycle, as practised by the majority of academics, but on foot. He always walked very briskly and, without having to unlock and lock a bike, arrived within the city no later than the others. Keeping up with him could prove a challenge whether walking around in Cambridge, along the river, to surrounding villages, or, as he liked to do, visiting the nearby stately homes of Anglesey Abbey, Wimpole Hall, and Audley End House. He had a flat in Hendon and enjoyed life there in London too, for example, up-market restaurants, or the theatre at which he always bought the best seats (as at the Footlights Theatre in Cambridge). The latter interest may be due to his cousin Heather Rechtman, who writes: 'We lived near Stratford East (London) where the fledgling company Theatre Workshop was just beginning to make a name for itself, and as a stage-struck teenager I dragged him along to many of their plays.' That was of course edgy stuff under Joan Littlewood, but he also enjoyed musicals. About one such trip to the theatre, Eva McLean narrates: '…we stopped over at his Hendon flat. While there, he decided to show me his mother's mink coat. When he opened the wardrobe door, hundreds of moths awoke to greet us. His response to this crisis? He shut the door firmly.' as well as: 'On another eventful outing to the West End, we arrived in good time and right there by the theatre came across a perfect parking place. Alan, however, decided it was too good to be true and so the search continued. When we next passed the spot, the parking space was gone — as was the first act of the play by the time we were inside.' Her funeral tribute (23 February 2018) sums up such anecdotes: 'So, in many ways, he was what Americans term "just a regular guy".' In his private life, he was quite relaxed, never going beyond sometimes stamping his foot when frustrated. However, in his professional life, he was often reserved, even insecure in some ways, and could on occasion be difficult. There were episodes, not just in Cambridge, which perhaps still have not been entirely forgotten. But it is now impossible to give any balanced accounts. Eva McLean also writes: 'His Jewish identity was also something which he strictly confined to the private sphere but was happy enough to discuss at length with co-religionists.I recollect an occasion when I told him about a particularly nasty anti-Semitic outpouring which I witnessed at a social event, adding how atypical that was these days. He, however, was convinced that anti-Semitism was never going away and was perhaps too sensitive to it. It must be remembered that when he first arrived in Cambridge (in 1961), church and chapel were as yet prominent in university life — a pillar of the academic establishment. He was worried about fitting in, contrasting his humble East End origins with the professional and even aristocratic backgrounds of many of the members of his college.Nevertheless, he still abided by the dietary rules forbidding the consumption of pork and shellfish and requiring meat to be kosher — ancient rules that even vast numbers of practising Jews regularly attending synagogue services conveniently overlook. Whenever we ate out he ordered fish, preferably smoked, but he found culinary life more restricted in College until, to his great relief, vegetarianism finally took hold. When in Hendon, all such requirements were, of course, easily met, and that is also where he would have shown up for services from time to time — foremost while his mother was still alive.All this should help solve a mystery that so baffled Trinity that his obituary in The Times opened with it — namely as to why he arrived for dinner every evening late by just the few minutes necessary to miss grace, especially when as the most senior Fellow he would have been obliged to read it. He just felt that he was not the right person for the task.' See 〈〈15〉〉 for the above-mentioned obituary. And Paula Tretkoff writes about a meeting, also with her mother: 'Alan asked to see our Israeli passports that had come up in conversation. He wanted to see if he could still read the Hebrew writing in them. He managed OK, though he no doubt only learned Hebrew as a child without pursuing it further.' Alan was enthusiastic about travel, which in America started already in 1969 with visiting professorships in Ann Arbor, Michigan, and Boulder, Colorado, and in 1970 membership of the Institute for Advanced Study in Princeton, New Jersey. He had at least three offers of chairs. One of these he turned down because the ivy-clad walls were too much like those in Cambridge, as was the weather. He preferred both modernity and sunshine. It was also thought that he wanted to look after his mother in London. (This writer recalls around 1980 seeing two people slowly crossing a Maryland street, and recognizing one, but not the other, and he followed them into a restaurant, where Alan introduced his mother.) He made many visits to America, and Eva McLean has shown me a lot of his postcards and other correspondence from abroad, for example, from Texas (1984), New York (1987), and California (1989). Of course, as his reputation grew he was able to widen his horizons, for example, to Australia, China and Hong Kong (missive 1988), India, Japan (postcard 1983), Russia, and many parts of Europe. In later life, he made regular trips to Switzerland (postcards 1989 and 1990) to work with Gisbert Wüstholz at ETH (a 1988 missive opens with 'greetings from a gnome of Zürich'). There were social events too, and he thought nothing of occasionally bringing presents such as a frying pan to dinner invitations — unconventional to be sure, but with its own unassailable logic. It was there, during a conference in honour of his 60th birthday, that he gave an entertaining and surprisingly candid speech about his life, starting with his recollections of wartime London, also mentioning that he was regarded as the 'brainbox' of the family, including one more rendering of Davenport's accent, and ending with his regrets about never marrying. Some people already knew that these regrets were not abstract but concerned specific ladies. I quote again from the funeral tribute: 'I first met Alan in 1975 playing table tennis in the Graduate Centre. I was immediately taken by his generosity of spirit. Unlike some of the others, he never minded being beaten. He proved to be equally generous and attentive as a suitor, and later as an old friend.' Alan's last years were made more difficult by increasing deafness and a series of falls (he did not pay serious attention to medical advice on these). He had long since advanced from Whewell's Court to Great Court and indeed had been proud to have the very best set of rooms there, like a maisonette on two floors. It overlooked the Bowling Green where he now could no longer play (see the cover of 〈〈13〉〉, the proceedings of the Zürich conference). Alan Baker died on 4 February 2018 in Cambridge after a severe stroke a few days earlier. During a Feast at Trinity given in his memory, he was described as 'idiosyncratic' in place of words like 'eccentric' or 'enigmatic'. Other biographical articles about Alan Baker can be found in the Hardy–Ramanujan Journal 〈〈1〉〉. See also 〈〈14〉〉 for a scientific appraisal by Wüstholz, which also contains a complete list of his publications. 2 Mathematics: a preview For a more structured narrative, we shall divide Alan Baker's work into eight categories (and in scientific detachment usually drop the first name). (a) Diophantine approximation. (b) Linear forms in logarithms. (c) Diophantine equations. (d) Elliptic functions. (e) Class numbers. (f) Abcology. (g) Miscellaneous. (h) Books. But before starting, we would like to mention perhaps the most easily stated of all his deep results. When we make a list of the perfect squares 1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 , 121 , 144 , … , 143384152921 , … , we see that the gaps between consecutive members get larger and larger (and in a regular way). Similarly for the perfect cubes 1 , 8 , 27 , 64 , 125 , 216 , 343 , 512 , 729 , 1000 , 1331 , 1728 , … , 143384152904 , … . But, if we mesh the two lists together to obtain the 'squbes' 1 , 4 , 8 , 9 , 16 , 25 , 27 , 36 , 49 , 64 , 81 , … , 143384152904 , 143384152921 , … , then it is not so clear that the gaps become large. Indeed this was proved by Mordell only in 1922; thus, for example, the gap 17 (twice above) or the gap 1621 (to take a year apparently at random) occurs at most finitely often. Unfortunately Mordell's proof gave no way of determining all the occurrences of a given gap. To do this amounts to specifying a non-zero k in the set Z of rational integers and finding all x , y in Z with y 2 = x 3 + k . (2.1)Baker achieved this in 1968 by showing that they all satisfy max { | x | , | y | } ⩽ exp { ( 10 10 | k | ) 10000 } . (2.2)Despite (2.1) being around since at least the year 1621 (and the cases k = − 2 , − 4 were set by Fermat in 1657 as a challenge to 'you English'), there were no estimates at all for x , y until (2.2) nearly 350 years later. Thus, for example, to find all gaps 1621, one just has to examine all y 2 and x 3 with y , x between 1 and say 10 10 132098 . This looks hopelessly impractical; yet we shall see later that Baker (with Davenport) found exceedingly efficient ways to do such things. There is an attractive single-sentence reformulation: for all positive integers x , y with x 3 ≠ y 2 , we have | x 3 − y 2 | > 10 − 10 ( log x ) 1 / 10000 . 3 Diophantine approximation It is classical that π = 3.1415926 … and 355 113 = 3.1415929 … are suspiciously close. This raises the natural question: given a real number ξ, how well can we approximate it by a rational number p / q ? An answer in convenient form was found by Dirichlet: provided ξ is not already rational, we can find infinitely many p / q with ξ − p q < 1 q 2 . (3.1) We pause to give the simple proof, which involves the Box Principle or Pigeonhole Principle. Pick any Q ⩾ 1 in Z. For i = 0 , 1 , … , Q we can find p i in Z with the Q + 1 pigeons θ i = i ξ − p i in the interval from 0 to 1. We divide this interval into Q holes of length 1 / Q . There are more pigeons than holes, so at least one hole must contain at least two pigeons. With say θ j and θ k ( j < k ), this leads at once to | q ξ − p | ⩽ 1 Q (3.2)for p = p k − p j and q = k − j satisfying 1 ⩽ q ⩽ Q . And now (3.1) follows; it is not too hard to see that we get infinitely many p / q as Q varies. But the answer to the next natural question of whether we can beat (3.1) depends critically on the number ξ. For a class of ξ particularly interesting to number-theorists, this topic can be said to have begun in earnest with Liouville in 1844, although it underlies the older concept of continued fractions such as 3 + 1 7 + 1 16 = 355 113 , or infinite ones like 3 + 1 7 + 1 15 + 1 1 + 1 292 + ⋯ = π . More generally, a 0 + 1 a 1 + 1 a 2 + 1 a 3 + 1 a 4 + ⋯ (3.3)is usually linearized to just x = [ a 0 ; a 1 , a 2 , a 3 , a 4 , … ] . (3.4)For example, α = [ 1 ; 1 , 1 , 1 , 1 , … ] = 1 + 5 2 , (3.5)and this is actually an algebraic number in that α 2 − α − 1 = 0 . On the other hand, Liouville's work (see later) shows that, if the positive integers a 0 , a 1 , a 2 , a 3 , a 4 , … increase very rapidly, then x in (3.4) does not satisfy any equation b 0 x d + b 1 x d − 1 + ⋯ + b d = 0 (3.6)for b 0 , b 1 , … , b d in Z not all zero. Thus by definition x is a transcendental number. In 1906, Maillet had given a different sort of transcendental continued fraction. A typical example is to take (3.5) and replace 1 by 2 in the positions k 1 , k 2 , … , where now these k 1 , k 2 , … increase very rapidly. In his very first paper [1] from 1962, Baker simplified and improved that work and also made the estimates more explicit. A consequence here is that it suffices to take k n as small as 4 n for transcendence. For the proof, one notes that something like x = [ 1 ; 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 2 , 1 , … ] is rather close to α above, with similar approximations further along. In this case, an easy generalization of Liouville is applicable, but for other examples it is necessary to use Klaus Roth's 'revolutionary improvement' (Cassels) of Liouville's result, or more precisely a consequence by Davenport and Roth, and even LeVeque's generalization of Roth. Acknowledgements such as Baker's: 'I should like to thank Professor Davenport for his valuable suggestions and help in preparing the manuscript.' can be seen quite often around this time in the journal Mathematika (which Davenport founded). The paper [4] from 1964 can be considered as a sort of continuation of [1]. To describe some of its results, we must recall that what Liouville proved is that, for any algebraic number α of degree d ⩾ 2 (the smallest integer such that α satisfies an equation (3.6) above), there is c > 0 such that α − p q ⩾ c q d (3.7)for all p and q ⩾ 1 in Z. When d ⩾ 3 , Roth improved this to α − p q ⩾ c q κ (3.8)for any κ > 2 , where now c is allowed to depend on κ. The Box Principle as in (3.1) shows that this is essentially best possible. Equivalently, if, for some ξ and κ > 2 , there is an infinite sequence of p i and q i ⩾ 1 in Z with p i / q i different and ξ − p i q i < 1 q i κ ( i = 1 , 2 , … ) , (3.9)then ξ must be transcendental. Baker then shows under an additional condition that ξ cannot be too close to an algebraic number in the sense of what is since called a 'transcendence measure'. The condition is that there should exist λ with q i + 1 ⩽ q i λ for all i. In that case, he shows that, for each n, there is μ n (possibly depending on ξ) such that | h 0 ξ n + h 1 ξ n − 1 + ⋯ + h n | > H − μ n (3.10)for all h 0 , h 1 , … , h n in Z not all zero, where H = max { 2 , | h 0 | , | h 1 | , … , | h n | } . The significance of this is the following. From (3.10) by definition (due to Kurt Mahler), ξ cannot be a so-called U-number, even though (3.9) implies that the 'partial quotients' a 0 , a 1 , … in (3.4) for x = ξ are unbounded. On the other hand, the partial quotients for sums of the form ∑ k = 1 ∞ 2 − k ! are also unbounded and it is a U-number (for example, (3.10) fails already for n = 1 and h 0 = 2 k ! with k large). Furthermore, Baker shows that the method of [1] produces U-numbers with bounded partial quotients, and that a suitable generalization of the above result to quadratic fields produces ξ that are not U-numbers, but still have bounded partial quotients. More succinctly, there is no correlation between the properties of having bounded partial quotients and being U-numbers. In fact, Baker is able to sharpen this to involve (also in Mahler's classification) T-numbers and S-numbers (see just below), and that was his main motivation. A spin-off is that either T-numbers exist or S-numbers 'of type exceeding 1' exist. This foreshadows Wolfgang Schmidt's breakthrough four years later showing that T-numbers exist. The proofs are rather formidable; indeed (3.9) and (3.10) lie close to a strengthening of Roth's Theorem, and accordingly Baker has to ramp up Roth's entire machinery, even in the situation of LeVeque's generalization. This time he writes only: 'I am indebted to Professor Davenport for valuable suggestions in connection with the present work.' In [15] from 1967, Baker returned to these themes, with generalizations to several numbers, at least in the case of bounded partial quotients. This property for a single ξ is equivalent to | ξ − p / q | ⩾ c / q 2 analogous to (3.7), and the natural extension to a pair ( ξ 1 , ξ 2 ) is max ξ 1 − p 1 q , ξ 2 − p 2 q ⩾ c q 3 / 2 for all p 1 , p 2 and q ⩾ 1 in Z. He shows, for example, that there are U-numbers ξ 1 , ξ 2 satisfying this, and also ξ 1 , ξ 2 that are not U-numbers. This time the proofs need no Roth-type considerations but follow a still intricate 'interval nesting' technique of Cassels and Davenport. In 1966, Baker [10] investigated certain 'metrical' properties of S-numbers. These are defined by refining (3.10) as follows. There is an ω such that for each n and each κ > n ω , we can find c > 0 (possibly depending on ξ , n , κ ) such that | h 0 ξ n + h 1 ξ n − 1 + ⋯ + h n | > c H − κ , (3.11)as in (3.10). For simplicity, we consider only real ξ. Much as before, the Box Principle shows that ω ⩾ 1 . It had been conjectured by Mahler in 1932 that we can in fact take ω = 1 ('of type exactly 1') for almost all ξ in the sense of Lebesgue measure. This was proved by Sprindzhuk in 1965. Having seen only preliminary announcements of that result, Baker was able to refine (3.11) even further to things like | h 0 ξ n + h 1 ξ n − 1 + ⋯ + h n | > c H − n ( log H ) − λ (3.12)for any λ > n . Later on in 1970, he considered with Schmidt [29] further refinements in terms of Hausdorff dimension; these are closer to Koksma's classification into S ∗ -, T ∗ -, and U ∗ -numbers based instead on the distance from ξ to algebraic numbers. Many of the results above can be expressed more concisely through the notation ∥ x ∥ = min m ∈ Z | x − m | for x in the field R of real numbers, meaning the distance to the nearest integer. For example, (3.7) says that ∥ q α ∥ ⩾ c / q d − 1 for all q ⩾ 1 . This notation can be used in other contexts. Returning to 1964, we may cite Baker's nice note [3], which shows that | Q | ∥ Q Θ 1 ∥ ∥ Q Θ 2 ∥ ⩾ e − 5 (3.13)for any non-zero polynomial Q in say R [ t ] , where Θ 1 = e 1 / t and Θ 2 = e 2 / t (same e, but that is irrelevant) are interpreted as formal power series in the topological completion with respect to the valuation | t | = e , and ∥ X ∥ = min M ∈ R [ t ] | X − M | measures the distance to the nearest polynomial. The significance here is that the analogue of (3.13) with R [ t ] replaced by Z (see just below) is thought to be false, by a famous conjecture of Littlewood (still unsolved). The note makes explicit an earlier result of Davenport and Lewis and provides a simpler proof, essentially by differentiating and using a Padé approximation (see later in this section). In fact, the Padé element can be eliminated to give a yet simpler proof as follows. It can be checked that Δ = Q P 1 P 2 t 2 Q ′ t 2 P 1 ′ + P 1 t 2 P 2 ′ + 2 P 2 t 4 Q ′ ′ + 2 t 3 Q ′ t 4 P 1 ′ ′ + ( 2 t 3 + 2 t 2 ) P 1 ′ + P 1 t 4 P 2 ′ ′ + ( 2 t 3 + 4 t 2 ) P 2 ′ + 4 P 2 remains unchanged on replacing P 1 , P 2 by E 1 = P 1 − Q e 1 / t and E 2 = P 2 − Q e 2 / t , respectively. Now suppose that Q , P 1 , P 2 , even in C [ t ] , are all non-zero. Inspecting the coefficients of smallest powers of t in each entry shows that Δ ≠ 0 . Thus | Δ | ⩾ 1 . Next, for Q ≠ 0 , choose P 1 , P 2 so that | E 1 | = ∥ Q e 1 / t ∥ < 1 and | E 2 | = ∥ Q e 2 / t ∥ < 1 . Clearly, P 1 ≠ 0 , P 2 ≠ 0 . Using | X ′ | ⩽ e − 1 | X | , we easily obtain that 1 ⩽ | Δ | ⩽ e 3 | Q | | E 1 | | E 2 | giving a slight improvement of the result. He evidently had this note in mind when writing [7] in 1965, back in R. A special result there is that, for any ε > 0 , there is c > 0 such that q ∥ q θ 1 ∥ ∥ q θ 2 ∥ ⩾ c q ε (3.14)for any q ⩾ 1 in Z, where θ 1 = e , θ 2 = e 2 . More generally Baker treats a product of several terms (as indeed in [3]) involving various numbers θ = e φ with (different) rational φ (that then implies quite easily the transcendence of e with quite a good measure); and he i