***This is a comment at the comments of Gerhard "Still Computing Oh So Slowly" Paseman's answer, giving just indexes n for more record-holders.***
Let $h_n$ denote the n'th harmonic number, $\small A_n=\{h_n\}$ its fractional part. The sequence of $A_n$ has a remarkable shape like a sawtooth-curve with sharp local minima when *n* is increased which can be exploited when we seek for possible *n* to be included in ***M***.
I use $\small w_n = \{h_n\} \cdot n$ and $\small f(n,\varepsilon)=w_n \cdot n^\varepsilon $ with the condition $\small f(n,\varepsilon)<1 $ as criterion for the inclusion of *n* into the set ***M*** . Of course the cardinality of ***M*** is limited by an upper bound $ \small n \le N$ with some ***N*** that can numerically be handled. I could manage to use $\small N \approx e^{2000}$ within Pari/GP.
The local minima of $\small A_n$ occur near $\small x_k=e^{k-\gamma}, k \in \mathbb N^+$ and the *n* to be tested is one of the next integers enclosing $\small x_k$, so this must be empirically be determined. (Note that my *n* are the *n+1* used in the comments at the OP denoting the previous *high* value of ***A*** )
The harmonic numbers can be computed in Pari/GP using `h(n) = psi(1+n) + Euler` ; however, this seems to be limited to something like $ \small n \lt e^{600}$ and so I had to introduce the Euler-McLaurin-formula for the larger $\small n \gt 1e50$
<hr>
The following table shows the first 40 entries of my 2000-row table using Pari/GP in a couple of seconds.
The basic option is now to test $\small w_n \cdot n^\varepsilon \lt 1$, for an example with $\small \varepsilon=0.1$ see the 6'th column and the 7'th column allowing to sum to the cardinality of ***M*** where we find $\small | M(0.1,N) | = 7$ with $\small n \le N \approx e^{2000}$
A second, much nicer, option is to define the function $\small e(n) = -\log_n(w_n) $ and test $\small e(n)>\varepsilon$ whether to include this *n* or not. This is simply possible looking into the 5'th column. This latter method allows to compute the cardinalities of $\small M(\varepsilon)$ for arbitrary $\varepsilon$ really fast, for instance to create informative scatter- or lineplots.
n | h_n | A=frac(h_n) | w=A*n | e(n) | w*n^0.1|in M?
-----------------+--------+-------------+----------+-----------+--------+---
1 1.00000 1.00000 1.00000 1.000000 1.00000 1
4 2.08333 0.0833333 0.333333 0.792481 0.382899 1
11 3.01988 0.0198773 0.218651 0.634006 0.277901 1
31 4.02725 0.0272452 0.844601 0.0491822 1.19066 .
83 5.00207 0.00206827 0.171667 0.398793 0.267051 1
227 6.00437 0.00436671 0.991243 0.00162136 1.70523 .
616 7.00127 0.00127410 0.784844 0.0377178 1.49191 .
1674 8.00049 0.000485572 0.812848 0.0279149 1.70759 .
4550 9.00021 0.000208063 0.946686 0.00650460 2.19790 .
12367 10.0000 0.0000430083 0.531883 0.0670005 1.36472 .
33617 11.0000 0.0000177086 0.595311 0.0497632 1.68811 .
91380 12.0000 0.00000305167 0.278861 0.111798 0.873923 1
248397 13.0000 0.00000122948 0.305399 0.0954806 1.05775 .
675214 14.0000 1.36205E-7 0.919678 0.00623806 3.52030 .
1835421 15.0000 3.78268E-7 0.694281 0.0252988 2.93703 .
4989191 16.0000 9.54538E-7 0.476237 0.0481002 2.22652 .
13562027 17.0000 1.48499E-8 0.201395 0.0975770 1.04059 .
36865412 18.0000 3.71993E-9 0.137137 0.114033 0.783098 1
100210581 19.0000 9.73330E-9 0.975380 0.00135314 6.15552 .
272400600 20.0000 1.61744E-9 0.440592 0.0421997 3.07297 .
740461601 21.0000 4.01333E-9 0.297172 0.0594162 2.29065 .
2012783315 22.0000 1.38447E-10 0.278664 0.0596444 2.37389 .
5471312310 23.0000 1.97920E-11 0.108288 0.0991384 1.01951 .
14872568831 24.0000 2.27220E-11 0.337935 0.0463183 3.51618 .
40427833596 25.0000 6.07937E-12 0.245776 0.0574601 2.82623 .
109894245429 26.0000 7.60776E-12 0.836049 0.00704359 10.6250 .
298723530401 27.0000 1.82203E-12 0.544283 0.0230213 7.64454 .
812014744422 28.0000 5.52830E-13 0.448906 0.0292071 6.96806 .
2207284924203 29.0000 1.00870E-13 0.222650 0.0528504 3.81951 .
6000022499693 30.0000 2.16954E-14 0.130173 0.0692963 2.46795 .
16309752131262 31.0000 3.65111E-14 0.595487 0.0170391 12.4772 .
44334502845080 32.0000 1.81005E-15 0.080247 0.0802804 1.85827 .
120513673457548 33.0000 4.59281E-15 0.553496 0.0182434 14.1651 .
327590128640500 34.0000 2.31992E-15 0.759983 0.00821173 21.4950 .
890482293866031 35.0000 2.71425E-17 0.024169 0.108145 0.755506 1
2420581837980561 36.0000 2.55560E-16 0.618603 0.0135588 21.3700 .
6579823624480555 37.0000 1.20561E-17 0.079326 0.0695767 3.02860 .
17885814992891026 38.0000 4.26642E-18 0.076308 0.0687541 3.21976 .
48618685882356024 39.0000 7.23781E-19 0.035189 0.0871101 1.64093 .
132159290357566703 40.0000 2.02186E-18 0.267208 0.0334763 13.7708 .
The set $M(0.1)$ is here the set of all $n$ where $\small w_n \cdot n^{0.1} \lt 1$ . The number of such entries for $\small n=132159290357566703 \approx 10^{17}$ is here *7* .
It might be interesting to look at logarithms of the included *n* . They are
ln(n) ln(n) + gamma
-------------------------------
0.0 0.57721
1.38629 1.96351
2.39789 2.97511
4.41884 4.99605
11.4227 12.0000
17.4227 18.0000
34.4227 35.0000
This - and the differences - might be interesting for extrapolating a tendency and a guess for the cardinality of $\small M(\varepsilon)$ and the range of *n* being checked.
Anyway, all numerical test which I've done indicate that the cardinality of $M$ is finite and even decreases with increasing $\varepsilon$ and only if $\varepsilon=0$ might be infinite.
<hr>
# Pari/GP tools
This is the Pari/GP-program which I used. Note, that the quotient of two consecutive $n$ approaches $\small \exp(1)$ so I need only the following guess:
Using the functions
h(n) = Euler+if(n<1e50, psi(1+n),log(n)+1/2/n-1/12/n^2+1/120/n^4-1/252/n^6+1/240/n^8)
A(n) = frac(h(n))
w(n) = A(n)*n
e(n) = if(n==1,return(1)); -log(w(n))/log(n)
The following needs only two steps to find the *n* with the next local minimum when called with index $\small k \in \mathbb N$ :
{find_n(k)=local(n1,n2,w1,w2);
n1=floor(exp(k-Euler)); n2=n1+1; w1 = A(n1);w2 = A(n2);
if(w1<w2,return(n1),return(n2));
}
\\ create list --------------------
{makeList(listlen=40)= local(list,n,w1,logn);
list=matrix(listlen,3);
list[1,]=[Euler,1,0,1];
for(k=2,listlen,
n=find_n(k); logn=log(n);
list[k,]=[logn+Euler,n,w1=w(n),-log(w1)/logn];
);
return(list); }
\\ compute cardinality with some eps
cardM(eps,list) = sum(k=1,#list[,1],list[k,4]>eps)
\\ apply, note: for long lists we need high internal precision
list = makeList(40)
print(cardM(0.1,list) )
<hr>
***[Update]***
This is new data for the OP's plot of the cardinality of $M$ by the argument $\varepsilon$ where $\small n <= \lceil(\exp(250-\gamma) \rceil$ :
eps | c=#M | r=c*eps | c= card(M) for n<= approx 10^108
------+------------------
0.01 104 1.040
0.02 55 1.100
0.03 36 1.080
0.04 29 1.160
0.05 21 1.050
0.06 16 0.960
0.07 12 0.840
0.08 12 0.960
0.09 10 0.900
0.10 7 0.700
0.11 6 0.660
0.12 4 0.480
0.13 4 0.520
0.14 4 0.560
0.15 4 0.600
0.16 4 0.640
0.17 4 0.680
0.18 4 0.720
0.19 4 0.760
0.20 4 0.800
<hr>
***[update 2]***
I've extended the search-space for $n$ to the range $\small 1 \ldots e^{1000} \approx 10^{434} $ and to check sanity to the range $\small 1 \ldots e^{2000} \approx 10^{868} $. For epsilons $\varepsilon$ from $0.001$ in $200$ steps up to $0.2$ I made the following graph:
[![picture][1]][1]
*(Sorry, I had changed the notation of $\small w(n)$ to $\small f(n)$ in that images and formulae)*
For the very small epsilon the increase of the search-space gives slightly higher results, which also illustrates, that for "larger" epsilon the cardinality of $\small M(\varepsilon)$ is finite
***[update 3]*** Uniformity of the $\small f(n) = frac(h_n) \cdot n^1$ at the *n*, where $\small A(n) $ has a local minimum (on request of @GerhardPaseman):
[![picture 2][2]][2]
It is also convenient to show a rescaling of the $\small f(n)$ so that we can immediately determine the cardinalities $\small |M(\varepsilon)|$ just by counting the number of dots above $\small e(n)\ge \varepsilon$. The derivation of the formula is
$$ \begin{array}{lll} f(n) \cdot n^\varepsilon &\lt & 1\\
\ln(f(n)) + \varepsilon \cdot \ln(n) &\lt& 0 \\
{\ln(f(n)) \over \ln(n)} + \varepsilon & \lt& 0 \\
\varepsilon & \lt& -\ln_n(f(n)) \end{array}$$
and we simply count, how many dots in the picture show $\small e(n) = - \ln_n(f(n)) \gt \varepsilon$
[![picture][3]][3]
<hr>
# Data for cardinalities
epsilon |M(eps,n)| |M(eps,n)|
n<=e^1000 n<=e^2000
-------------------------------
0.0000 1000 2000
0.0010 633 862
0.0020 430 482
0.0030 315 330
0.0040 258 264
0.0050 204 205
0.0060 171 171
0.0070 151 151
0.0080 135 135
0.0090 120 120
0.0100 109 109
0.0110 100 100
0.0120 98 98
0.0130 89 89
0.0140 84 84
0.0150 79 79
0.0160 73 73
0.0170 69 69
0.0180 62 62
0.0190 57 57
0.0200 55 55
0.0210 50 50
0.0220 47 47
0.0230 44 44
0.0240 42 42
0.0250 41 41
0.0260 39 39
0.0270 39 39
0.0280 37 37
0.0290 37 37
0.0300 36 36
0.0310 36 36
0.0320 36 36
0.0330 36 36
0.0340 34 34
0.0350 34 34
0.0360 34 34
0.0370 33 33
0.0380 32 32
0.0390 31 31
0.0400 29 29
0.0410 29 29
0.0420 29 29
0.0430 26 26
0.0440 26 26
0.0450 25 25
0.0460 25 25
0.0470 24 24
0.0480 24 24
0.0490 23 23
0.0500 21 21
0.0510 21 21
0.0520 21 21
0.0530 20 20
0.0540 20 20
0.0550 20 20
0.0560 20 20
0.0570 19 19
0.0580 18 18
0.0590 18 18
0.0600 16 16
0.0610 16 16
0.0620 16 16
0.0630 16 16
0.0640 16 16
0.0650 16 16
0.0660 16 16
0.0670 16 16
0.0680 15 15
0.0690 14 14
0.0700 12 12
0.0710 12 12
0.0720 12 12
0.0730 12 12
0.0740 12 12
0.0750 12 12
0.0760 12 12
0.0770 12 12
0.0780 12 12
0.0790 12 12
0.0800 12 12
0.0810 11 11
0.0820 11 11
0.0830 11 11
0.0840 11 11
0.0850 11 11
0.0860 11 11
0.0870 11 11
0.0880 10 10
0.0890 10 10
0.0900 10 10
0.0910 10 10
0.0920 10 10
0.0930 10 10
0.0940 10 10
0.0950 10 10
0.0960 9 9
0.0970 9 9
0.0980 8 8
0.0990 8 8
0.1000 7 7
0.1010 7 7
0.1020 7 7
0.1030 7 7
0.1040 7 7
0.1050 7 7
0.1060 7 7
0.1070 7 7
0.1080 7 7
0.1090 6 6
0.1100 6 6
0.1110 6 6
0.1120 5 5
0.1130 5 5
0.1140 5 5
0.1150 4 4
0.1160 4 4
0.1170 4 4
0.1180 4 4
0.1190 4 4
0.1200 4 4
0.1210 4 4
0.1220 4 4
0.1230 4 4
0.1240 4 4
0.1250 4 4
0.1260 4 4
0.1270 4 4
0.1280 4 4
0.1290 4 4
0.1300 4 4
0.1310 4 4
0.1320 4 4
0.1330 4 4
0.1340 4 4
0.1350 4 4
0.1360 4 4
0.1370 4 4
0.1380 4 4
0.1390 4 4
0.1400 4 4
0.1410 4 4
0.1420 4 4
0.1430 4 4
0.1440 4 4
0.1450 4 4
0.1460 4 4
0.1470 4 4
0.1480 4 4
0.1490 4 4
0.1500 4 4
0.1510 4 4
0.1520 4 4
0.1530 4 4
0.1540 4 4
0.1550 4 4
0.1560 4 4
0.1570 4 4
0.1580 4 4
0.1590 4 4
0.1600 4 4
0.1610 4 4
0.1620 4 4
0.1630 4 4
0.1640 4 4
0.1650 4 4
0.1660 4 4
0.1670 4 4
0.1680 4 4
0.1690 4 4
0.1700 4 4
0.1710 4 4
0.1720 4 4
0.1730 4 4
0.1740 4 4
0.1750 4 4
0.1760 4 4
0.1770 4 4
0.1780 4 4
0.1790 4 4
0.1800 4 4
0.1810 4 4
0.1820 4 4
0.1830 4 4
0.1840 4 4
0.1850 4 4
0.1860 4 4
0.1870 4 4
0.1880 4 4
0.1890 4 4
0.1900 4 4
0.1910 4 4
0.1920 4 4
0.1930 4 4
0.1940 4 4
0.1950 4 4
0.1960 4 4
0.1970 4 4
0.1980 4 4
0.1990 4 4
[1]: https://i.sstatic.net/DAz5g.png
[2]: https://i.sstatic.net/18Az3.png
[3]: https://i.sstatic.net/QFKcS.png